Chapter 3

Indefinite Integrals

[563/98%]

Show

LaTeX

Show LaTeX

Section 3.1The Simplest Indefinite Integrals[243/98%]

Problem 1628

Using the table of basic integrals, find the following integral: $\int (3-x^{2})^{3} dx$ .

Problem 1629

Using the table of basic integrals, find the following integral: $\int x^{2}(5-x)^{4} dx$ .

Problem 1630

Using the table of basic integrals, find the following integral: $\int (1-x)(1-2x)(1-3x) dx$ .

Problem 1631

Using the table of elementary integrals, find the following integral: $\int \left(\dfrac {1-x}{x}\right)^{2} dx$ .

Problem 1632

Using the table of elementary integrals, find the following integral: $\int \left(\dfrac {a}{x}+\dfrac {a^{2}}{x^{2}}+\dfrac {a^{3}}{x^{3}}\right) dx$ .

Problem 1633

Using the table of elementary integrals, find the following integral: $\int \dfrac {x+1}{\sqrt{x}} dx$ .

Problem 1634

Using the table of elementary integrals, find the following integral: $\int \dfrac {\sqrt{x}-2\sqrt[3]{x^{2}}+1}{\sqrt[4]{x}} dx$ .

Problem 1635

Using the table of elementary integrals, find the following integral: $\int \dfrac {(1-x)^{3}}{x\sqrt[3]{x}} dx$ .

Problem 1636

Using the table of elementary integrals, find the following integral: $\int \left(1-\dfrac {1}{x^{2}}\right)\sqrt{x\sqrt{x}} dx$ .

Problem 1637

Using the table of elementary integrals, find the following integral: $\int \dfrac {(\sqrt{2x}-\sqrt[3]{3x})^{2}}{x} dx$ .

Problem 1638

Using the table of elementary integrals, find the following integral: $\int \dfrac {\sqrt{x^{4}+x^{-4}+2}}{x^{3}} dx$ .

Problem 1639

Using the table of elementary integrals, find the following integral: $\int \dfrac {x^{2}}{1+x^{2}} dx$ .

Problem 1640

Using the table of elementary integrals, find the following integral: $\int \dfrac {x^{2}}{1-x^{2}} dx$ .

Problem 1641

Using the table of elementary integrals, find the following integral: $\int \dfrac {x^{2}+3}{x^{2}-1} dx$ .

Problem 1642

Using the table of elementary integrals, find the following integral: $\int \dfrac {\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1-x^{4}}} dx$ .

Problem 1643

Using the table of elementary integrals, find the following integral: $\int \dfrac {\sqrt{x^{2}+1}-\sqrt{x^{2}-1}}{\sqrt{x^{4}-1}} dx$ .

Problem 1644

Using the table of elementary integrals, find the following integral: $\int (2^{x}+3^{x})^{2} dx$ .

Problem 1645

Using the table of elementary integrals, find the following integral: $\int \dfrac {2^{x+1}-5^{x-1}}{10^{x}} dx$ .

Problem 1646

Using the table of elementary integrals, find the following integral: $\int \dfrac {e^{3x}+1}{e^{x}+1} dx$ .

Problem 1647

Using the table of elementary integrals, find the following integral: $\int (1+\sin x+\cos x) dx$ .

Problem 1648

Using the table of elementary integrals, find the following integral: $\int \sqrt{1-\sin 2x} dx$ .

Problem 1649

Using the table of elementary integrals, find the following integral: $\int \operatorname {ctg}^{2}x\, dx$ .

Problem 1650

Using the table of elementary integrals, find the following integral: $\int \operatorname {tg}^{2}x\, dx$ .

Problem 1651

Using the table of elementary integrals, find the following integral: $\int (a\operatorname {sh}x+b\operatorname {ch}x) dx$ .

Problem 1652

Using the table of elementary integrals, find the following integral: $\int \operatorname {th}^{2}x\, dx$ .

Problem 1653

Using the table of elementary integrals, find the following integral: $\int \operatorname {cth}^{2}x\, dx$ .

Problem 1654

Prove that if

\int f(x) dx=F(x)+C,

then

\int f(ax+b) dx=\frac{1}{a}F(ax+b)+C \quad (a\neq 0) .

Problem 1655

Using the table of elementary integrals, find the following integral: $\int \dfrac {dx}{x+a}$ .

Problem 1656

Using the table of elementary integrals, find the following integral: $\int (2x-3)^{10} dx$ .

Problem 1657

Using the table of elementary integrals, find the following integral: $\int \sqrt[3]{1-3x} dx$ .

Problem 1658

Using the table of elementary integrals, find the following integral: $\int \dfrac {dx}{\sqrt{2-5x}}$ .

Problem 1659

Using the table of elementary integrals, find the following integral: $\int \dfrac {dx}{(5x-2)^{\frac{5}{2}}}$ .

Problem 1660

Using the table of elementary integrals, find the following integral: $\int \dfrac {\sqrt[5]{1-2x+x^{2}}}{1-x} dx$ .

Problem 1661

Using the table of elementary integrals, find the following integral: $\int \dfrac {dx}{2+3x^{2}}$ .

Problem 1662

Using the table of elementary integrals, find the following integral: $\int \dfrac {dx}{2-3x^{2}}$ .

Problem 1663

Using the table of elementary integrals, find the following integral: $\int \dfrac {dx}{\sqrt{2-3x^{2}}}$ .

Problem 1664

Using the table of elementary integrals, find the following integral: $\int \dfrac {dx}{\sqrt{3x^{2}-2}}$ .

Problem 1665

Using the table of elementary integrals, find the following integral: $\int (e^{-x}+e^{-2x}) dx$ .

Problem 1666

Using the table of elementary integrals, find the following integral: $\int (\sin 5x-\sin 5a) dx$ .

Problem 1667

Using the table of elementary integrals, find the following integral: $\int \dfrac {dx}{\sin^{2}\left(2x+\dfrac {\pi }{4}\right)}$ .

Problem 1668

Using a table of basic integrals, find the following integral: $\int \dfrac {dx}{1+\cos x}$ .

Problem 1669

Using a table of basic integrals, find the following integral: $\int \dfrac {dx}{1-\cos x}$ .

Problem 1670

Using a table of basic integrals, find the following integral: $\int \dfrac {dx}{1+\sin x}$ .

Problem 1671

Using a table of basic integrals, find the following integral: $\int \left[\operatorname {sh}(2x+1)+\operatorname {ch}(2x-1)\right] dx$ .

Problem 1672

Using a table of basic integrals, find the following integral: $\int \dfrac {dx}{\operatorname {ch}^{2}\dfrac {x}{2}}$ .

Problem 1673

Using a table of basic integrals, find the following integral: $\int \dfrac {dx}{\operatorname {sh}^{2}\dfrac {x}{2}}$ .

Problem 1674

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {x\, dx}{\sqrt{1-x^{2}}}$ .

Problem 1675

Using a suitable transformation of the integrand, find the following integral: $\int x^{2}\sqrt[3]{1+x^{3}}\, dx$ .

Problem 1676

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {x\, dx}{3-2x^{2}}$ .

Problem 1677

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {x\, dx}{(1+x^{2})^{2}}$ .

Problem 1678

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {x\, dx}{4+x^{4}}$ .

Problem 1679

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {x^{3} dx}{x^{8}-2}$ .

Problem 1680

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{\sqrt{x}(1+x)}$ .

Problem 1681

Using a suitable transformation of the integrand, find the following integral: $\int \sin \dfrac {1}{x}\cdot \dfrac {dx}{x^{2}}$ .

Problem 1682

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{x\sqrt{x^{2}+1}}$ .

Problem 1683

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{x\sqrt{x^{2}-1}}$ .

Problem 1684

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{(x^{2}+1)^{\frac{3}{2}}}$ .

Problem 1685

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {x\, dx}{(x^{2}-1)^{\frac{3}{2}}}$ .

Problem 1686

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {x^{2} dx}{(8x^{3}+27)^{\frac{2}{3}}}$ .

Problem 1687

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{\sqrt{x(1+x)}}$ .

Problem 1688

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{\sqrt{x(1-x)}}$ .

Problem 1689

Using a suitable transformation of the integrand, find the following integral: $\int xe^{-x^{2}} dx$ .

Problem 1690

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {e^{x} dx}{2+e^{x}}$ .

Problem 1691

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{e^{x}+e^{-x}}$ .

Problem 1692

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{\sqrt{1+e^{2x}}}$ .

Problem 1693

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {\ln^{2}x}{x} dx$ .

Problem 1694

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{x\ln x\ln (\ln x)}$ .

Problem 1695

Using a suitable transformation of the integrand, find the following integral: $\int \sin^{5}x\cos x\, dx$ .

Problem 1696

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {\sin x}{\sqrt{\cos^{3}x}} dx$ .

Problem 1697

Using a suitable transformation of the integrand, find the following integral: $\int \operatorname {tg}x\, dx$ .

Problem 1698

Using a suitable transformation of the integrand, find the following integral: $\int \operatorname {ctg}x\, dx$ .

Problem 1699

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}} dx$ .

Problem 1700

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {\sin x\cos x}{\sqrt{a^{2}\sin^{2}x+b^{2}\cos^{2}x}} dx$ .

Problem 1700.1

[NO TASK]

Problem 1700.1

[NO TASK]

Problem 1700.2

[NO TASK]

Problem 1700.2

[NO TASK]

Problem 1700.3

[NO TASK]

Problem 1700.3

[NO TASK]

Problem 1701

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{\sin^{2}x\sqrt[4]{\operatorname {ctg}x}}$ .

Problem 1702

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{\sin^{2}x+2\cos^{2}x}$ .

Problem 1703

Using a suitable transformation of the integrand, find the following integral: $\int \dfrac {dx}{\sin x}$ .

Problem 1704

By suitably transforming the integrand, find the following integral: $\int \dfrac {dx}{\cos x}$ .

Problem 1705

By suitably transforming the integrand, find the following integral: $\int \dfrac {dx}{\operatorname {sh}x}$ .

Problem 1706

By suitably transforming the integrand, find the following integral: $\int \dfrac {dx}{\operatorname {ch}x}$ .

Problem 1707

By suitably transforming the integrand, find the following integral: $\int \dfrac {\operatorname {sh}x\operatorname {ch}x}{\sqrt{\operatorname {sh}^{4}x+\operatorname {ch}^{4}x}} dx$ .

Problem 1708

By suitably transforming the integrand, find the following integral: $\int \dfrac {dx}{\operatorname {ch}^{2}x\cdot \sqrt[3]{\operatorname {th}^{2}x}}$ .

Problem 1709

By suitably transforming the integrand, find the following integral: $\int \dfrac {\operatorname {arctg}x}{1+x^{2}} dx$ .

Problem 1710

By suitably transforming the integrand, find the following integral: $\int \dfrac {dx}{(\arcsin x)^{2}\sqrt{1-x^{2}}}$ .

Problem 1711

By suitably transforming the integrand, find the following integral: $\int \sqrt{\dfrac {\ln (x+\sqrt{1+x^{2}})}{1+x^{2}}} dx$ .

Problem 1712

By suitably transforming the integrand, find the following integral: $\int \dfrac {x^{2}+1}{x^{4}+1} dx$ .

Problem 1713

By suitably transforming the integrand, find the following integral: $\int \dfrac {x^{2}-1}{x^{4}+1} dx$ .

Problem 1714

By suitably transforming the integrand, find the following integral: $\int \dfrac {x^{14} dx}{(x^{5}+1)^{4}}$ .

Problem 1715

By suitably transforming the integrand, find the following integral: $\int \dfrac {x^{\frac{n}{2}} dx}{\sqrt{1+x^{n+2}}}$ .

Problem 1716

By suitably transforming the integrand, find the following integral: $\int \dfrac {1}{1-x^{2}}\ln \dfrac {1+x}{1-x} dx$ .

Problem 1717

By suitably transforming the integrand, find the following integral: $\int \dfrac {\cos x\, dx}{\sqrt{2+\cos 2x}}$ .

Problem 1718

By suitably transforming the integrand, find the following integral: $\int \dfrac {\sin x\cos x}{\sin^{4}x+\cos^{4}x} dx$ .

Problem 1719

By suitably transforming the integrand, find the following integral: $\int \dfrac {2^{x}\cdot 3^{x}}{9^{x}-4^{x}} dx$ .

Problem 1720

By suitably transforming the integrand, find the following integral: $\int \dfrac {x\, dx}{\sqrt{1+x^{2}}+\sqrt{(1+x^{2})^{3}}}$ .

Problem 1721

Using term-by-term integration, find the following integral: $\int x^{2}(2-3x^{2})^{2} dx$ .

Problem 1721.1

[NO TASK]

Problem 1721.1

[NO TASK]

Problem 1722

Using term-by-term integration, find the following integral: $\int \dfrac {1+x}{1-x} dx$ .

Problem 1723

Using term-by-term integration, find the following integral: $\int \dfrac {x^{2}}{1+x} dx$ .

Problem 1724

Using term-by-term integration, find the following integral: $\int \dfrac {x^{3}}{3+x} dx$ .

Problem 1725

Using term-by-term integration, find the following integral: $\int \dfrac {(1+x)^{2}}{1+x^{2}} dx$ .

Problem 1726

Using term-by-term integration, find the following integral: $\int \dfrac {(2-x)^{2}}{2-x^{2}} dx$ .

Problem 1727

Using term-by-term integration, find the following integral: $\int \dfrac {x^{2}}{(1-x)^{100}} dx$ .

Problem 1728

Using term-by-term integration, find the following integral: $\int \dfrac {x^{5}}{x+1} dx$ .

Problem 1729

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{\sqrt{x+1}+\sqrt{x-1}}$ .

Problem 1730

Using term-by-term integration, find the following integral: $\int x\sqrt{2-5x}\, dx$ .

Problem 1731

Using term-by-term integration, find the following integral: $\int \dfrac {x\, dx}{\sqrt[3]{1-3x}}$ .

Problem 1732

Using term-by-term integration, find the following integral: $\int x^{3}\sqrt[3]{1+x^{2}}\, dx$ .

Problem 1733

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{(x-1)(x+3)}$ .

Problem 1734

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{x^{2}+x-2}$ .

Problem 1735

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{(x^{2}+1)(x^{2}+2)}$ .

Problem 1736

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{(x^{2}-2)(x^{2}+3)}$ .

Problem 1737

Using term-by-term integration, find the following integral: $\int \dfrac {x\, dx}{(x+2)(x+3)}$ .

Problem 1738

Using term-by-term integration, find the following integral: $\int \dfrac {x\, dx}{x^{4}+3x^{2}+2}$ .

Problem 1739

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{(x+a)^{2}(x+b)^{2}} \quad (a\neq b)$ .

Problem 1740

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{(x^{2}+a^{2})(x^{2}+b^{2})} \quad (\left|a\right|\neq \left|b\right|)$ .

Problem 1741

Using term-by-term integration, find the following integral: $\int \sin^{2}x\, dx$ .

Problem 1742

Using term-by-term integration, find the following integral: $\int \cos^{2}x\, dx$ .

Problem 1743

Using term-by-term integration, find the following integral: $\int \sin x\cdot \sin (x+a)\, dx$ .

Problem 1744

Using term-by-term integration, find the following integral: $\int \sin 3x\cdot \sin 5x\, dx$ .

Problem 1745

Using term-by-term integration, find the following integral: $\int \cos \dfrac {x}{2}\cdot \cos \dfrac {x}{3} dx$ .

Problem 1746

Using term-by-term integration, find the following integral: $\int \sin \left(2x-\dfrac {\pi }{6}\right)\cdot \cos \left(3x+\dfrac {\pi }{4}\right) dx$ .

Problem 1747

Using term-by-term integration, find the following integral: $\int \sin^{3}x\, dx$ .

Problem 1748

Using term-by-term integration, find the following integral: $\int \cos^{3}x\, dx$ .

Problem 1749

Using term-by-term integration, find the following integral: $\int \sin^{4}x\, dx$ .

Problem 1750

Using term-by-term integration, find the following integral: $\int \cos^{4}x\, dx$ .

Problem 1751

Using term-by-term integration, find the following integral: $\int \operatorname {ctg}^{2}x\, dx$ .

Problem 1752

Using term-by-term integration, find the following integral: $\int \operatorname {tg}^{3}x\, dx$ .

Problem 1753

Using term-by-term integration, find the following integral: $\int \sin^{2}3x\cdot \sin^{3}2x\, dx$ .

Problem 1754

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{\sin^{2}x\cdot \cos^{2}x}$ .

Problem 1755

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{\sin^{2}x\cdot \cos x}$ .

Problem 1756

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{\sin x\cdot \cos^{3}x}$ .

Problem 1757

Using term-by-term integration, find the following integral: $\int \dfrac {\cos^{3}x}{\sin x} dx$ .

Problem 1758

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{\cos^{4}x}$ .

Problem 1759

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{1+e^{x}}$ .

Problem 1760

Using term-by-term integration, find the following integral: $\int \dfrac {(1+e^{x})^{2}}{1+e^{2x}} dx$ .

Problem 1761

Using term-by-term integration, find the following integral: $\int \operatorname {sh}^{2}x\, dx$ .

Problem 1762

Using term-by-term integration, find the following integral: $\int \operatorname {ch}^{2}x\, dx$ .

Problem 1763

Using term-by-term integration, find the following integral: $\int \operatorname {sh}x\cdot \operatorname {sh}2x\, dx$ .

Problem 1764

Using term-by-term integration, find the following integral: $\int \operatorname {ch}x\cdot \operatorname {ch}3x\, dx$ .

Problem 1765

Using term-by-term integration, find the following integral: $\int \dfrac {dx}{\operatorname {sh}^{2}x\cdot \operatorname {ch}^{2}x}$ .

Problem 1766

Using a suitable substitution, find the following integral: $\int x^{2}\sqrt[3]{1-x}\, dx$ .

Problem 1767

Using a suitable substitution, find the following integral: $\int x^{3}(1-5x^{2})^{10} dx$ .

Problem 1768

Using a suitable substitution, find the following integral: $\int \dfrac {x^{2}}{\sqrt{2-x}} dx$ .

Problem 1769

Using a suitable substitution, find the following integral: $\int \dfrac {x^{5}}{\sqrt{1-x^{2}}} dx$ .

Problem 1770

Using a suitable substitution, find the following integral: $\int x^{5}(2-5x^{3})^{\frac{2}{3}} dx$ .

Problem 1771

Using a suitable substitution, find the following integral: $\int \cos^{5}x\sqrt{\sin x}\, dx$ .

Problem 1772

Using a suitable substitution, find the following integral: $\int \dfrac {\sin x\cos^{3}x}{1+\cos^{2}x} dx$ .

Problem 1773

Using a suitable substitution, find the following integral: $\int \dfrac {\sin^{2}x}{\cos^{6}x} dx$ .

Problem 1774

Using a suitable substitution, find the following integral: $\int \dfrac {\ln x\, dx}{x\sqrt{1+\ln x}}$ .

Problem 1775

Using a suitable substitution, find the following integral: $\int \dfrac {dx}{e^{\frac{x}{2}}+e^{x}}$ .

Problem 1776

Using a suitable substitution, find the following integral: $\int \dfrac {dx}{\sqrt{1+e^{x}}}$ .

Problem 1777

Using a suitable substitution, find the following integral: $\int \dfrac {\operatorname {arctg}\sqrt{x}}{\sqrt{x}}\cdot \dfrac {dx}{1+x}$ .

Problem 1778

Using a trigonometric substitution $x=a\sin t$ , $x=a\operatorname {tg}t$ , $x=a\sin^{2}t$ , etc. (with positive parameter), find the following integral: $\int \dfrac {dx}{(1-x^{2})^{\frac{3}{2}}}$ .

Problem 1779

Using a trigonometric substitution $x=a\sin t$ , $x=a\operatorname {tg}t$ , $x=a\sin^{2}t$ , etc. (with positive parameter), find the following integral: $\int \dfrac {x^{2} dx}{\sqrt{x^{2}-2}}$ .

Problem 1780

Using a trigonometric substitution $x=a\sin t$ , $x=a\operatorname {tg}t$ , $x=a\sin^{2}t$ , etc. (with positive parameter), find the following integral: $\int \sqrt{a^{2}-x^{2}}\, dx$ .

Problem 1781

Using a trigonometric substitution $x=a\sin t$ , $x=a\operatorname {tg}t$ , $x=a\sin^{2}t$ , etc. (with positive parameter), find the following integral: $\int \dfrac {dx}{(x^{2}+a^{2})^{\frac{3}{2}}}$ .

Problem 1782

Using a trigonometric substitution $x=a\sin t$ , $x=a\operatorname {tg}t$ , $x=a\sin^{2}t$ , etc. (with positive parameter), find the following integral: $\int \sqrt{\dfrac {a+x}{a-x}}\, dx$ .

Problem 1783

Using a trigonometric substitution $x=a\sin t$ , $x=a\operatorname {tg}t$ , $x=a\sin^{2}t$ , etc. (with positive parameter), find the following integral: $\int x\sqrt{\dfrac {x}{2a-x}}\, dx$ .

Problem 1784

Using a trigonometric substitution $x=a\sin t$ , $x=a\operatorname {tg}t$ , $x=a\sin^{2}t$ , etc. (with positive parameter), find the following integral: $\int \dfrac {dx}{\sqrt{(x-a)(b-x)}}$ .

Problem 1785

Using a trigonometric substitution $x=a\sin t$ , $x=a\operatorname {tg}t$ , $x=a\sin^{2}t$ , etc. (with positive parameter), find the following integral: $\int \sqrt{(x-a)(b-x)}\, dx$ .

Problem 1786

Using a hyperbolic substitution $x=a\operatorname {sh}t$ , $x=a\operatorname {ch}t$ , etc. (with positive parameter), find the following integral: $\int \sqrt{a^{2}+x^{2}}\, dx$ .

Problem 1787

Using a hyperbolic substitution $x=a\operatorname {sh}t$ , $x=a\operatorname {ch}t$ , etc. (with positive parameter), find the following integral: $\int \dfrac {x^{2}}{\sqrt{a^{2}+x^{2}}} dx$ .

Problem 1788

Using a hyperbolic substitution $x=a\operatorname {sh}t$ , $x=a\operatorname {ch}t$ , etc. (with positive parameter), find the following integral: $\int \sqrt{\dfrac {x-a}{x+a}}\, dx$ .

Problem 1789

Using a hyperbolic substitution $x=a\operatorname {sh}t$ , $x=a\operatorname {ch}t$ , etc. (with positive parameter), find the following integral: $\int \dfrac {dx}{\sqrt{(x+a)(x+b)}}$ .

Problem 1790

Using a hyperbolic substitution $x=a\operatorname {sh}t$ , $x=a\operatorname {ch}t$ , etc. (with positive parameter), find the following integral: $\int \sqrt{(x+a)(x+b)}\, dx$ .

Problem 1791

Using integration by parts, find the following integral: $\int \ln x\, dx$ .

Problem 1792

Using integration by parts, find the following integral: $\int x^{n}\ln x\, dx \quad (n\neq -1)$ .

Problem 1793

Using integration by parts, find the following integral: $\int \left(\dfrac {\ln x}{x}\right)^{2} dx$ .

Problem 1794

Using integration by parts, find the following integral: $\int \sqrt{x}\ln^{2}x\, dx$ .

Problem 1795

Using integration by parts, find the following integral: $\int xe^{-x} dx$ .

Problem 1796

Using integration by parts, find the following integral: $\int x^{2}e^{-2x} dx$ .

Problem 1797

Using integration by parts, find the following integral: $\int x^{3}e^{-x^{2}} dx$ .

Problem 1798

Using integration by parts, find the following integral: $\int x\cos x\, dx$ .

Problem 1799

Using integration by parts, find the following integral: $\int x^{2}\sin 2x\, dx$ .

Problem 1800

Using integration by parts, find the following integral: $\int x\operatorname {sh}x\, dx$ .

Problem 1801

Using integration by parts, find the following integral: $\int x^{3}\operatorname {ch}3x\, dx$ .

Problem 1802

Using integration by parts, find the following integral: $\int \operatorname {arctg}x\, dx$ .

Problem 1803

Using integration by parts, find the following integral: $\int \arcsin x\, dx$ .

Problem 1804

Using integration by parts, find the following integral: $\int x\operatorname {arctg}x\, dx$ .

Problem 1805

Using integration by parts, find the following integral: $\int x^{2}\arccos x\, dx$ .

Problem 1806

Using integration by parts, find the following integral: $\int \dfrac {\arcsin x}{x^{2}} dx$ .

Problem 1807

Using integration by parts, find the following integral: $\int \ln (x+\sqrt{1+x^{2}})\, dx$ .

Problem 1808

Using integration by parts, find the following integral: $\int x\ln \dfrac {1+x}{1-x}\, dx$ .

Problem 1809

Using integration by parts, find the following integral: $\int \operatorname {arctg}\sqrt{x}\, dx$ .

Problem 1810

Using integration by parts, find the following integral: $\int \sin x\cdot \ln (\operatorname {tg}x)\, dx$ .

Problem 1811

Find the following integral: $\int x^{5}e^{x^{3}} dx$ .

Problem 1812

Find the following integral: $\int (\arcsin x)^{2} dx$ .

Problem 1813

Find the following integral: $\int x(\operatorname {arctg}x)^{2} dx$ .

Problem 1814

Find the following integral: $\int x^{2}\ln \dfrac {1-x}{1+x}\, dx$ .

Problem 1815

Find the following integral: $\int \dfrac {x\ln (x+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}} dx$ .

Problem 1816

Find the following integral: $\int \dfrac {x^{2}}{(1+x^{2})^{2}} dx$ .

Problem 1817

Find the following integral: $\int \dfrac {dx}{(a^{2}+x^{2})^{2}}$ .

Problem 1818

Find the following integral: $\int \sqrt{a^{2}-x^{2}}\, dx$ .

Problem 1819

Find the following integral: $\int \sqrt{x^{2}+a}\, dx$ .

Problem 1820

Find the following integral: $\int x^{2}\sqrt{a^{2}+x^{2}}\, dx$ .

Problem 1821

Find the following integral: $\int x\sin^{2}x\, dx$ .

Problem 1822

Find the following integral: $\int e^{\sqrt{x}} dx$ .

Problem 1823

Find the following integral: $\int x\sin \sqrt{x}\, dx$ .

Problem 1824

Find the following integral: $\int \dfrac {xe^{\operatorname {arctg}x}}{(1+x^{2})^{\frac{3}{2}}} dx$ .

Problem 1825

Find the following integral: $\int \dfrac {e^{\operatorname {arctg}x}}{(1+x^{2})^{\frac{3}{2}}} dx$ .

Problem 1826

Find the following integral: $\int \sin (\ln x)\, dx$ .

Problem 1827

Find the following integral: $\int \cos (\ln x)\, dx$ .

Problem 1828

Find the following integral: $\int e^{ax}\cos bx\, dx$ .

Problem 1829

Find the following integral: $\int e^{ax}\sin bx\, dx$ .

Problem 1830

Find the following integral: $\int e^{2x}\sin^{2}x\, dx$ .

Problem 1831

Find the following integral: $\int (e^{x}-\cos x)^{2} dx$ .

Problem 1832

Find the following integral: $\int \dfrac {\operatorname {arcctg}e^{x}}{e^{x}} dx$ .

Problem 1833

Find the following integral: $\int \dfrac {\ln (\sin x)}{\sin^{2}x} dx$ .

Problem 1834

Find the following integral: $\int \dfrac {x\, dx}{\cos^{2}x}$ .

Problem 1835

Find the following integral: $\int \dfrac {xe^{x}}{(x+1)^{2}} dx$ .

Problem 1836

The following integrals require completing the square to reduce the quadratic trinomial to standard form, and using the following formulas:

\begin{align} & \text{I.} \quad \int \dfrac {dx}{a^{2}+x^{2}}=\frac{1}{a}\operatorname {arctg}\frac{x}{a}+C \quad (a\neq 0); \\ & \text{II.} \quad \int \dfrac {dx}{a^{2}-x^{2}}=\frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right|+C \quad (a\neq 0); \\ & \text{III.} \quad \int \dfrac {x\, dx}{a^{2}\pm x^{2}}=\pm \frac{1}{2}\ln \left|a^{2}\pm x^{2}\right|+C; \\ & \text{IV.} \quad \int \dfrac {dx}{\sqrt{a^{2}-x^{2}}}=\arcsin \frac{x}{a}+C \quad (a>0); \\ & \text{V.} \quad \int \dfrac {dx}{\sqrt{x^{2}\pm a^{2}}}=\ln \left|x+\sqrt{x^{2}\pm a^{2}}\right|+C; \\ & \text{VI.} \quad \int \dfrac {x\, dx}{\sqrt{a^{2}\pm x^{2}}}=\pm \sqrt{a^{2}\pm x^{2}}+C; \\ & \text{VII.} \quad \int \sqrt{a^{2}-x^{2}}\, dx=\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}\arcsin \frac{x}{a}+C \quad (a>0); \\ & \text{VIII.} \quad \int \sqrt{x^{2}\pm a^{2}}\, dx=\frac{x}{2}\sqrt{x^{2}\pm a^{2}}\pm \frac{a^{2}}{2}\ln \left|x+\sqrt{x^{2}\pm a^{2}}\right|+C . \end{align}

Find the following integral: $\int \dfrac {dx}{a+bx^{2}} \quad (ab\neq 0)$ .

Problem 1837

Find the following integral: $\int \dfrac {dx}{x^{2}-x+2}$ .

Problem 1838

Find the following integral: $\int \dfrac {dx}{3x^{2}-2x-1}$ .

Problem 1839

Find the following integral: $\int \dfrac {x\, dx}{x^{4}-2x^{2}-1}$ .

Problem 1840

Find the following integral: $\int \dfrac {x+1}{x^{2}+x+1} dx$ .

Problem 1841

Find the following integral: $\int \dfrac {x\, dx}{x^{2}-2x\cos \alpha +1}$ .

Problem 1842

Find the following integral: $\int \dfrac {x^{3}\, dx}{x^{4}-x^{2}+2}$ .

Problem 1843

Find the following integral: $\int \dfrac {x^{5}\, dx}{x^{6}-x^{3}-2}$ .

Problem 1844

Find the following integral: $\int \dfrac {dx}{3\sin^{2}x-8\sin x\cos x+5\cos^{2}x}$ .

Problem 1845

Find the following integral: $\int \dfrac {dx}{\sin x+2\cos x+3}$ .

Problem 1846

Find the following integral: $\int \dfrac {dx}{\sqrt{a+bx^{2}}} \quad (b\neq 0)$ .

Problem 1847

Find the following integral: $\int \dfrac {dx}{\sqrt{1-2x-x^{2}}}$ .

Problem 1848

Find the following integral: $\int \dfrac {dx}{\sqrt{x+x^{2}}}$ .

Problem 1849

Find the following integral: $\int \dfrac {dx}{\sqrt{2x^{2}-x+2}}$ .

Problem 1850

Prove that if

y=ax^{2}+bx+c \quad (a\neq 0),

then when $a>0$ , $\displaystyle \int \dfrac {dx}{\sqrt{y}}=\dfrac {1}{\sqrt{a}}\ln \left|\dfrac {y’}{2}+\sqrt{ay}\right|+C$ ;

when $a<0$ , $\displaystyle \int \dfrac {dx}{\sqrt{y}}=\dfrac {1}{\sqrt{-a}}\arcsin \dfrac {-y’}{\sqrt{b^{2}-4ac}}+C$ .

Problem 1851

Find the following integral: $\int \dfrac {x\, dx}{\sqrt{5+x-x^{2}}}$ .

Problem 1852

Find the following integral: $\int \dfrac {x+1}{\sqrt{x^{2}+x+1}} dx$ .

Problem 1853

Find the following integral: $\int \dfrac {x\, dx}{\sqrt{1-3x^{2}-2x^{4}}}$ .

Problem 1853.1

[NO TASK]

Problem 1853.1

[NO TASK]

Problem 1854

Find the following integral: $\int \dfrac {x^{3}\, dx}{\sqrt{x^{4}-2x^{2}-1}}$ .

Problem 1855

Find the following integral: $\int \dfrac {x+x^{3}}{\sqrt{1+x^{2}-x^{4}}} dx$ .

Problem 1856

Find the following integral: $\int \dfrac {dx}{x\sqrt{x^{2}+x+1}}$ .

Problem 1857

Find the following integral: $\int \dfrac {dx}{x^{2}\sqrt{x^{2}+x-1}}$ .

Problem 1858

Find the following integral: $\int \dfrac {dx}{(x+1)\sqrt{x^{2}+1}}$ .

Problem 1859

Find the following integral: $\int \dfrac {dx}{(x-1)\sqrt{x^{2}-2}}$ .

Problem 1860

Find the following integral: $\int \dfrac {dx}{(x+2)^{2}\sqrt{x^{2}+2x-5}}$ .

Problem 1861

Find the following integral: $\int \sqrt{2+x-x^{2}}\, dx$ .

Problem 1862

Find the following integral: $\int \sqrt{2+x+x^{2}}\, dx$ .

Problem 1863

Find the following integral: $\int \sqrt{x^{4}+2x^{2}-1}\, x\, dx$ .

Problem 1864

Find the following integral: $\int \dfrac {1-x+x^{2}}{x\sqrt{1+x-x^{2}}} dx$ .

Problem 1865

Find the following integral: $\int \dfrac {x^{2}+1}{x\sqrt{x^{4}+1}} dx$ .

Section 3.2Integration of rational functions[60/100%]

Problem 1866

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {2x+3}{(x-2)(x+5)} dx$ .

Problem 1867

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x\, dx}{(x+1)(x+2)(x+3)}$ .

Problem 1868

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{10}}{x^{2}+x-2} dx$ .

Problem 1869

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{3}+1}{x^{3}-5x^{2}+6x} dx$ .

Problem 1870

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{4}}{x^{4}+5x^{2}+4} dx$ .

Problem 1871

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x\, dx}{x^{3}-3x+2}$ .

Problem 1872

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{2}+1}{(x+1)^{2}(x-1)} dx$ .

Problem 1873

Using the method of undetermined coefficients, find the following integral: $\int \left(\dfrac {x}{x^{2}-3x+2}\right)^{2} dx$ .

Problem 1874

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{(x+1)(x+2)^{2}(x+3)^{3}}$ .

Problem 1875

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{5}+x^{4}-2x^{3}-2x^{2}+x+1}$ .

Problem 1876

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{2}+5x+4}{x^{4}+5x^{2}+4} dx$ .

Problem 1877

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{(x+1)(x^{2}+1)}$ .

Problem 1878

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{(x^{2}-4x+4)(x^{2}-4x+5)}$ .

Problem 1879

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x\, dx}{(x-1)^{2}(x^{2}+2x+2)}$ .

Problem 1880

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x(1+x)(1+x+x^{2})}$ .

Problem 1881

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{3}+1}$ .

Problem 1882

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x\, dx}{x^{3}-1}$ .

Problem 1883

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{4}-1}$ .

Problem 1884

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{4}+1}$ .

Problem 1885

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{4}+x^{2}+1}$ .

Problem 1886

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{6}+1}$ .

Problem 1887

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{(1+x)(1+x^{2})(1+x^{3})}$ .

Problem 1888

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {dx}{x^{5}-x^{4}+x^{3}-x^{2}+x-1}$ .

Problem 1889

Using the method of undetermined coefficients, find the following integral: $\int \dfrac {x^{2}\, dx}{x^{4}+3x^{3}+\dfrac {9}{2}x^{2}+3x+1}$ .

Problem 1890

Under what condition is the integral $\displaystyle \int \dfrac {ax^{2}+bx+c}{x^{3}(x-1)^{2}} dx$ a rational function?

Problem 1891

Using the Ostrogradsky method, compute the integral: $\int \dfrac {x\, dx}{(x-1)^{2}(x+1)^{3}}$ .

Remark.

The so-called Ostrogradsky method refers to the following: for the integral of a proper rational fraction $\dfrac {P(x)}{Q(x)}$ , algebraic means can be used to split it into the sum of a proper fraction and the integral of another proper fraction, in such a way that in the new integrand's proper fraction the degree of the denominator is brought down to its lowest possible state, that is, in the formula

\int \dfrac {P(x)}{Q(x)} dx=\dfrac {P_{1}(x)}{Q_{1}(x)}+\int \dfrac {P_{2}(x)}{Q_{2}(x)} dx \tag {1}

where, if $P(x)$ , $Q(x)$ are known, and the denominator $Q(x)$ is supposed to factor into real linear and quadratic-type factors:

Q(x)=(x-a)^{k}\cdots (x^{2}+px+q)^{m}\cdots ,

where $k,\ldots ,m,\ldots$ are natural numbers. On the right-hand side of formula (1) the denominators are known, of the form:

\begin{align} Q_{1}(x) & =(x-a)^{k-1}\cdots (x^{2}+px+q)^{m-1}\cdots , \\ Q_{2}(x) & =(x-a)\cdots (x^{2}+px+q)\cdots , \end{align}

and satisfy $Q_{1}(x)\cdot Q_{2}(x)=Q(x)$ . Meanwhile $P_{1}(x)$ and $P_{2}(x)$ are polynomials of lower degree than $Q_{1}(x)$ and $Q_{2}(x)$ respectively, and can generally be found by the method of undetermined coefficients. This method of using formula (1) to find the integral is precisely the so-called Ostrogradsky method. For details, see G.M. Fikhtengolts (Peking University translation), A Course of Differential and Integral Calculus, Volume 2, Part 1, p.264. --- Note of the solutions editor.

Problem 1892

Using the Ostrogradsky method, compute the integral: $\int \dfrac {dx}{(x^{3}+1)^{2}}$ .

Problem 1893

Using the Ostrogradsky method, compute the integral: $\int \dfrac {dx}{(x^{2}+1)^{3}}$ .

Problem 1894

Using the Ostrogradsky method, compute the integral: $\int \dfrac {x^{2}\, dx}{(x^{2}+2x+2)^{2}}$ .

Problem 1895

Using the Ostrogradsky method, compute the integral: $\int \dfrac {dx}{(x^{4}+1)^{2}}$ .

Problem 1896

Using the Ostrogradsky method, compute the integral: $\displaystyle \int \dfrac {x^{2}+3x-2}{(x-1)(x^{2}+x+1)^{2}} dx$ .

Problem 1897

Using the Ostrogradsky method, compute the integral: $\int \dfrac {dx}{(x^{4}-1)^{3}}$ .

Problem 1898

Extract the algebraic part of the following integral: $\displaystyle \int \dfrac {x^{2}+1}{(x^{4}+x^{2}+1)^{2}} dx$ .

Problem 1899

Extract the algebraic part of the following integral: $\displaystyle \int \dfrac {dx}{(x^{3}+x+1)^{3}}$ .

Problem 1900

Extract the algebraic part of the following integral: $\displaystyle \int \dfrac {4x^{5}-1}{(x^{5}+x+1)^{2}} dx$ .

Problem 1901

Compute the integral: $\displaystyle \int \dfrac {dx}{x^{4}+2x^{3}+3x^{2}+2x+1}$ .

Problem 1902

Under what condition is the integral $\displaystyle \int \dfrac {\alpha x^{2}+2\beta x+\gamma }{(ax^{2}+2bx+c)^{2}} dx$ a rational function?

Problem 1903

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{3}}{(x-1)^{100}} dx$ .

Problem 1904

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x\, dx}{x^{8}-1}$ .

Problem 1905

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{3}\, dx}{x^{8}+3}$ .

Problem 1906

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{2}+x}{x^{6}+1} dx$ .

Problem 1907

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{4}-3}{x(x^{8}+3x^{4}+2)} dx$ .

Problem 1908

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{4}\, dx}{(x^{10}-10)^{2}}$ .

Problem 1909

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{11}\, dx}{x^{8}+3x^{4}+2}$ .

Problem 1910

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{9}\, dx}{(x^{10}+2x^{5}+2)^{2}}$ .

Problem 1911

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{2n-1}}{x^{n}+1} dx$ .

Problem 1912

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{3n-1}}{(x^{2n}+1)^{2}} dx$ .

Problem 1913

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {dx}{x(x^{10}+2)}$ .

Problem 1914

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {dx}{x(x^{10}+1)^{2}}$ .

Problem 1915

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {1-x^{7}}{x(1+x^{7})} dx$ .

Problem 1916

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{4}-1}{x(x^{4}-5)(x^{5}-5x+1)} dx$ .

Problem 1917

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{2}+1}{x^{4}+x^{2}+1} dx$ .

Problem 1918

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{2}-1}{x^{4}+x^{3}+x^{2}+x+1} dx$ .

Problem 1919

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{5}-x}{x^{8}+1} dx$ .

Problem 1920

Using various methods, calculate the following integral: $\displaystyle \int \dfrac {x^{4}+1}{x^{6}+1} dx$ .

Problem 1921

Derive a reduction formula for computing the integral $\displaystyle I_{n}=\int \dfrac {dx}{(ax^{2}+bx+c)^{n}} (a\neq 0)$ .

Using this formula, compute $\displaystyle I_{3}=\int \dfrac {dx}{(x^{2}+x+1)^{3}}$ .

Problem 1922

Using the substitution $t=\dfrac {x+a}{x+b}$ , compute the integral:

I=\int \dfrac {dx}{(x+a)^{m}(x+b)^{n}}

( $m$ and $n$ are natural numbers).

Using this substitution, find $\displaystyle \int \dfrac {dx}{(x-2)^{2}(x+3)^{3}}$ .

Problem 1923

If $P_{n}(x)$ is a polynomial of degree $n$ in $x$ , compute $\displaystyle \int \dfrac {P_{n}(x)}{(x-a)^{n+1}} dx$ .

Problem 1924

Let $R(x)=R^{*}(x^{2})$ , where $R^{*}$ is a rational function. What special properties does the function $R(x)$ have when it is decomposed into partial fractions?

Problem 1925

Compute $\displaystyle \int \dfrac {dx}{1+x^{2n}}$ , where $n$ is a positive integer.

Section 3.3Integration of Irrational Functions[65/100%]

Problem 1926

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {dx}{1+\sqrt{x}}$ .

Problem 1927

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {dx}{x(1+2\sqrt{x}+\sqrt[3]{x})}$ .

Problem 1928

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {x\sqrt[3]{2+x}}{x+\sqrt[3]{2+x}} dx$ .

Problem 1929

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {1-\sqrt{x+1}}{1+\sqrt[3]{x+1}} dx$ .

Problem 1930

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {dx}{\sqrt{x}(1+\sqrt[4]{x})^{3}}$ .

Problem 1931

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}} dx$ .

Problem 1932

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {dx}{\sqrt[3]{(x+1)^{2}(x-1)^{4}}}$ .

Problem 1933

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {x\, dx}{\sqrt[4]{x^{3}(a-x)}} (a>0)$ .

Problem 1934

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {dx}{\sqrt[n]{(x-a)^{n+1}(x-b)^{n-1}}}$ ( $n$ is a natural number).

Problem 1935

Convert the integrand to a rational function to find the following integral: $\displaystyle \int \dfrac {dx}{1+\sqrt{x}+\sqrt{1+x}}$ .

Problem 1936

Prove: if $p+q=kn$ , where $k$ is an integer, then the integral

\int R\left[x, (x-a)^{\frac{p}{n}}(x-b)^{\frac{q}{n}}\right] dx

(where $R$ is a rational function and $p$ , $q$ , $n$ are integers) is an elementary function.

Problem 1937

Find the integral of the simplest quadratic irrational expression: $\displaystyle \int \dfrac {x^{2}}{\sqrt{1+x+x^{2}}} dx$ .

Problem 1938

Convert the integrand into a rational function in order to find the following integral: $\displaystyle \int \dfrac {dx}{(1+x)\sqrt{x^{2}+x+1}}$ .

Problem 1939

Find the integral of the simplest quadratic irrational expression: $\displaystyle \int \dfrac {dx}{(1-x)^{2}\sqrt{1-x^{2}}}$ .

Problem 1940

Find the integral of the simplest quadratic irrational expression: $\displaystyle \int \dfrac {\sqrt{x^{2}+2x+2}}{x} dx$ .

Problem 1941

Find the integral of the simplest quadratic irrational expression: $\displaystyle \int \dfrac {x\, dx}{(1+x)\sqrt{1-x-x^{2}}}$ .

Problem 1942

Find the integral of the simplest quadratic irrational expression: $\displaystyle \int \dfrac {1-x+x^{2}}{\sqrt{1+x-x^{2}}} dx$ .

Problem 1943

Using the formula

\int \dfrac {P_{n}(x)}{y} dx=Q_{n-1}(x)y+\lambda \int \dfrac {dx}{y} ，

where $y=\sqrt{ax^{2}+bx+c}$ , $P_{n}(x)$ is a polynomial of degree $n$ , $Q_{n-1}(x)$ is a polynomial of degree $n-1$ , and $\lambda$ is a constant, find the following integral: $\displaystyle \int \dfrac {x^{3}}{\sqrt{1+2x-x^{2}}} dx$ .

Problem 1944

Using the formula

\int \dfrac {P_{n}(x)}{y} dx=Q_{n-1}(x)y+\lambda \int \dfrac {dx}{y} ，

Problem 1945

Using the formula

\int \dfrac {P_{n}(x)}{y} dx=Q_{n-1}(x)y+\lambda \int \dfrac {dx}{y} ，

Problem 1946

Using the formula

\int \dfrac {P_{n}(x)}{y} dx=Q_{n-1}(x)y+\lambda \int \dfrac {dx}{y} ，

Problem 1947

Find the integral: $\displaystyle \int \dfrac {dx}{x^{3}\sqrt{x^{2}+1}}$ .

Problem 1948

Find the integral: $\displaystyle \int \dfrac {dx}{x^{4}\sqrt{x^{2}-1}}$ .

Problem 1949

Find the integral: $\displaystyle \int \dfrac {dx}{(x-1)^{3}\sqrt{x^{2}+3x+1}}$ .

Problem 1950

Find the integral: $\displaystyle \int \dfrac {dx}{(x+1)^{5}\sqrt{x^{2}+2x}}$ .

Problem 1951

Under what conditions is the integral $\displaystyle \int \dfrac {a_{1}x^{2}+b_{1}x+c_{1}}{\sqrt{ax^{2}+bx+c}} dx$ an algebraic function?

Problem 1952

Find the integral: $\displaystyle \int \dfrac {x\, dx}{(x-1)^{2}\sqrt{1+2x-x^{2}}}$ .

Problem 1953

Find the integral: $\displaystyle \int \dfrac {x\, dx}{(x^{2}-1)\sqrt{x^{2}-x-1}}$ .

Problem 1954

Find the integral: $\displaystyle \int \dfrac {\sqrt{x^{2}+x+1}}{(x+1)^{2}} dx$ .

Problem 1955

Find the integral: $\displaystyle \int \dfrac {x^{3}}{(1+x)\sqrt{1+2x-x^{2}}} dx$ .

Problem 1956

Find the integral: $\displaystyle \int \dfrac {x\, dx}{(x^{2}-3x+2)\sqrt{x^{2}-4x+3}}$ .

Problem 1957

Find the integral: $\displaystyle \int \dfrac {dx}{(1+x^{2})\sqrt{1-x^{2}}}$ .

Problem 1958

Find the integral: $\displaystyle \int \dfrac {dx}{(x^{2}+1)\sqrt{x^{2}-1}}$ .

Problem 1959

Find the integral: $\displaystyle \int \dfrac {dx}{(1-x^{4})\sqrt{1+x^{2}}}$ .

Problem 1960

Find the integral: $\displaystyle \int \dfrac {\sqrt{x^{2}+2}}{x^{2}+1} dx$ .

Problem 1961

Reduce the quadratic trinomial to canonical form in order to compute the following integral: $\displaystyle \int \dfrac {dx}{(x^{2}+x+1)\sqrt{x^{2}+x-1}}$ .

Problem 1962

Reduce the quadratic trinomial to canonical form in order to compute the following integral: $\displaystyle \int \dfrac {x^{2}\, dx}{(4-2x+x^{2})\sqrt{2+2x-x^{2}}}$ .

Problem 1963

Find the integral: $\displaystyle \int \dfrac {(x+1)\, dx}{(x^{2}+x+1)\sqrt{x^{2}+x+1}}$ .

Problem 1964

Using the linear fractional substitution $x=\dfrac {\alpha +\beta t}{1+t}$ , compute the integral: $\displaystyle \int \dfrac {dx}{(x^{2}-x+1)\sqrt{x^{2}+x+1}}$ .

Problem 1965

Find $\displaystyle \int \dfrac {dx}{(x^{2}+2)\sqrt{2x^{2}-2x+5}}$ .

Problem 1966

Using Euler's substitutions

if $a>0$ , $\sqrt{ax^{2}+bx+c}=\pm \sqrt{a}x+z$ ;
if $c>0$ , $\sqrt{ax^{2}+bx+c}=xz\pm \sqrt{c}$ ;
$\sqrt{a(x-x_{1})(x-x_{2})}=z(x-x_{1})$ , find the following integral: $\displaystyle \int \dfrac {dx}{x+\sqrt{x^{2}+x+1}}$ .

Problem 1967

Find the integral: $\displaystyle \int \dfrac {dx}{1+\sqrt{1-2x-x^{2}}}$ .

Problem 1968

Find the integral: $\displaystyle \int x\sqrt{x^{2}-2x+2}\, dx$ .

Problem 1969

Find the integral: $\displaystyle \int \dfrac {x-\sqrt{x^{2}-3x+2}}{x+\sqrt{x^{2}+3x+2}}\, dx$ .

Problem 1970

Find the integral: $\displaystyle \int \dfrac {dx}{\left[1+\sqrt{x(1+x)}\right]^{2}}$ .

Problem 1971

Using various methods, compute the following integral: $\displaystyle \int \dfrac {dx}{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}}$ .

Problem 1972

Find the integral: $\displaystyle \int \dfrac {x\, dx}{(1-x^{3})\sqrt{1-x^{2}}}$ .

Problem 1973

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt{2}+\sqrt{1-x}+\sqrt{1+x}}$ .

Problem 1974

Find the integral: $\displaystyle \int \dfrac {x+\sqrt{1+x+x^{2}}}{1+x+\sqrt{1+x+x^{2}}}\, dx$ .

Problem 1975

Find the integral: $\displaystyle \int \dfrac {\sqrt{x(x+1)}}{\sqrt{x}+\sqrt{x+1}}\, dx$ .

Problem 1976

Find the integral: $\displaystyle \int \dfrac {(x^{2}-1)\, dx}{(x^{2}+1)\sqrt{x^{4}+1}}$ .

Problem 1977

Find the integral: $\displaystyle \int \dfrac {x^{2}+1}{(x^{2}-1)\sqrt{x^{4}+1}}\, dx$ .

Problem 1978

Find the integral: $\displaystyle \int \dfrac {dx}{x\sqrt{x^{4}+2x^{2}-1}}$ .

Problem 1979

Find the integral: $\displaystyle \int \dfrac {(x^{2}+1)\, dx}{x\sqrt{x^{4}+x^{2}+1}}$ .

Problem 1980

Prove that the method for finding the integral $\displaystyle \int R\left(x,\sqrt{ax+b},\sqrt{cx+d}\right)\, dx$ (where $R$ is a rational function) reduces to the integration of a rational function.

Problem 1981

The binomial differential $\displaystyle \int x^{m}(a+bx^{n})^{p}\, dx$ (where $m$ , $n$ , $p$ are rational numbers) can be reduced to the integral of a rational function only in the following three cases (Chebyshev's theorem):

Case one, $p$ is an integer. Set $x=z^{N}$ , where $N$ is the common denominator of the fractions $m$ and $n$ .

Case two, $\dfrac {m+1}{n}$ is an integer. Set $a+bx^{n}=z^{N}$ , where $N$ is the denominator of the fraction $p$ .

Case three, $\dfrac {m+1}{n}+p$ is an integer. Use the substitution $ax^{-n}+b=z^{N}$ , where $N$ is the denominator of the fraction $p$ .

Compute the following integral: $\displaystyle \int \sqrt{x^{3}+x^{4}}\, dx$ .

Problem 1982

Find the integral: $\displaystyle \int \dfrac {\sqrt{x}}{(1+\sqrt[3]{x})^{2}}\, dx$ .

Problem 1983

Find the integral: $\displaystyle \int \dfrac {x\, dx}{\sqrt{1+\sqrt[3]{x^{2}}}}$ .

Problem 1984

Find the integral: $\displaystyle \int \dfrac {x^{5}\, dx}{\sqrt{1-x^{2}}}$ .

Problem 1985

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt[3]{1+x^{3}}}$ .

Problem 1986

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt[4]{1+x^{4}}}$ .

Problem 1987

Find the integral: $\displaystyle \int \dfrac {dx}{x\sqrt[6]{1+x^{6}}}$ .

Problem 1988

Find the integral: $\displaystyle \int \dfrac {dx}{x^{3}\sqrt[5]{1+\dfrac {1}{x}}}$ .

Problem 1989

Find the integral: $\displaystyle \int \sqrt[3]{3x-x^{3}}\, dx$ .

Problem 1990

Under what conditions is the integral $\displaystyle \int \sqrt{1+x^{m}}\, dx$ (where $m$ is a rational number) an elementary function?

Section 3.4Integration of trigonometric functions[76/99%]

Problem 1991

Compute using the recursion formula. Find the following integral: $\displaystyle \int \cos^{5}x\, dx$ .

Problem 1992

Find the integral: $\displaystyle \int \sin^{6}x\, dx$ .

Problem 1993

Find the integral: $\displaystyle \int \cos^{6}x\, dx$ .

Problem 1994

Find the integral: $\displaystyle \int \sin^{2}x\cos^{4}x\, dx$ .

Problem 1995

Find the integral: $\displaystyle \int \sin^{4}x\cos^{5}x\, dx$ .

Problem 1996

Find the integral: $\displaystyle \int \sin^{5}x\cos^{5}x\, dx$ .

Problem 1997

Find the integral: $\displaystyle \int \dfrac {\sin^{3}x}{\cos^{4}x}\, dx$ .

Problem 1998

Find the integral: $\displaystyle \int \dfrac {\cos^{4}x}{\sin^{3}x}\, dx$ .

Problem 1999

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{3}x}$ .

Problem 2000

Find the integral: $\displaystyle \int \dfrac {dx}{\cos^{3}x}$ .

Problem 2001

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{4}x\cos^{4}x}$ .

Problem 2002

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{3}x\cos^{5}x}$ .

Problem 2003

Find the integral: $\displaystyle \int \dfrac {dx}{\sin x\cos^{4}x}$ .

Problem 2004

Find the integral: $\displaystyle \int \operatorname {tg}^{5}x\, dx$ .

Problem 2005

Find the integral: $\displaystyle \int \operatorname {ctg}^{6}x\, dx$ .

Problem 2006

Find the integral: $\displaystyle \int \dfrac {\sin^{4}x}{\cos^{8}x}\, dx$ .

Problem 2007

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt{\sin^{3}x\cos^{5}x}}$ .

Problem 2008

Find the integral: $\displaystyle \int \dfrac {dx}{\cos x\sqrt[3]{\sin^{2}x}}$ .

Problem 2009

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt{\operatorname {tg}x}}$ .

Problem 2010

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt[3]{\operatorname {tg}x}}$ .

Problem 2011

(a)(b)

Derive recursion formulas for the following integrals:

(1) $I_{n}=\displaystyle \int \sin^{n}x\, dx$ ；

(2) $K_{n}=\displaystyle \int \cos^{n}x\, dx$ ( $n>2$ ). and use the resulting formulas to compute $\displaystyle \int \sin^{6}x\, dx$ and $\displaystyle \int \cos^{8}x\, dx$ .

(a)

(b)

Problem 2012

(a)(b)

Derive recursion formulas for the following integrals:

(1) $I_{n}=\displaystyle \int \dfrac {dx}{\sin^{n}x}$ ；

(2) $K_{n}=\displaystyle \int \dfrac {dx}{\cos^{n}x}$ ( $n>2$ ). and use the resulting formulas to compute $\displaystyle \int \dfrac {dx}{\sin^{5}x}$ and $\displaystyle \int \dfrac {dx}{\cos^{7}x}$ .

(a)

(b)

Problem 2013

Using the formulas

\text{I}\quad \sin \alpha \sin \beta =\frac{1}{2}\left[\cos (\alpha -\beta )-\cos (\alpha +\beta )\right] ;

\text{II}\quad \cos \alpha \cos \beta =\frac{1}{2}\left[\cos (\alpha +\beta )+\cos (\alpha -\beta )\right] ;

\text{III}\quad \sin \alpha \cos \beta =\frac{1}{2}\left[\sin (\alpha -\beta )+\sin (\alpha +\beta )\right]

to compute the following integrals. Find the integral: $\displaystyle \int \sin 5x\cos x\, dx$ .

Problem 2014

Find the integral: $\displaystyle \int \cos x\cos 2x\cos 3x\, dx$ .

Problem 2015

Find the integral: $\displaystyle \int \sin x\sin \dfrac {x}{2}\sin \dfrac {x}{3}\, dx$ .

Problem 2016

Find the integral: $\displaystyle \int \sin x\sin (x+a)\sin (x+b)\, dx$ .

Problem 2017

Find the integral: $\displaystyle \int \cos^{2}ax\cos^{2}bx\, dx$ .

Problem 2018

Find the integral: $\displaystyle \int \sin^{3}2x\cos^{2}3x\, dx$ .

Problem 2019

Find the integral: $\displaystyle \int \dfrac {dx}{\sin (x+a)\sin (x+b)}$ .

Problem 2020

Find the integral: $\displaystyle \int \dfrac {dx}{\sin (x+a)\cos (x+b)}$ .

Problem 2021

Find the integral: $\displaystyle \int \dfrac {dx}{\cos (x+a)\cos (x+b)}$ .

Problem 2022

Find the integral: $\displaystyle \int \dfrac {dx}{\sin x-\sin a}$ .

Problem 2023

Find the integral: $\displaystyle \int \dfrac {dx}{\cos x+\cos a}$ .

Problem 2024

Find the integral: $\displaystyle \int \operatorname {tg}x\operatorname {tg}(x+a)\, dx$ .

Problem 2025

(a)(b)

For an integral of the form $\displaystyle \int R(\sin x,\cos x)\, dx$ (where $R$ is a rational function), the general case can be reduced to the integral of a rational function by the substitution $\operatorname {tg}\dfrac {x}{2}=t$ .

(a)

If the identity $R(-\sin x,\cos x)\equiv -R(\sin x,\cos x)$ holds, or $R(\sin x,-\cos x)\equiv -R(\sin x,\cos x)$ holds, then it is best to use the substitution $\cos x=t$ or, correspondingly, $\sin x=t$ .

(b)

If the identity $R(-\sin x,-\cos x)\equiv R(\sin x,\cos x)$ holds, then it is best to use the substitution $\operatorname {tg}x=t$ .

Find the integral: $\displaystyle \int \dfrac {dx}{2\sin x-\cos x+5}$ .

Problem 2026

Find the integral: $\displaystyle \int \dfrac {dx}{(2+\cos x)\sin x}$ .

Problem 2027

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x}{\sin x+2\cos x}\, dx$ .

Problem 2028

(a)(b)

Find the integral: $\displaystyle \int \dfrac {dx}{1+\varepsilon \cos x}$ :

(1) $0<\varepsilon <1$ ;

(2) $\varepsilon >1$ .

(a)

(b)

Problem 2029

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x}{1+\sin^{2}x}\, dx$ .

Problem 2030

Find the integral: $\displaystyle \int \dfrac {dx}{a^{2}\sin^{2}x+b^{2}\cos^{2}x}$ .

Problem 2031

Find the integral: $\displaystyle \int \dfrac {\cos^{2}x\, dx}{(a^{2}\sin^{2}x+b^{2}\cos^{2}x)^{2}}$ .

Problem 2032

Find the integral: $\displaystyle \int \dfrac {\sin x\cos x}{\sin x+\cos x}\, dx$ .

Problem 2033

Find the integral: $\displaystyle \int \dfrac {dx}{(a\sin x+b\cos x)^{2}}$ .

Problem 2034

Find the integral: $\displaystyle \int \dfrac {\sin x\, dx}{\sin^{3}x+\cos^{3}x}$ .

Problem 2035

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{4}x+\cos^{4}x}$ .

Problem 2036

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x\cos^{2}x}{\sin^{8}x+\cos^{8}x}\, dx$ .

Problem 2037

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x-\cos^{2}x}{\sin^{4}x+\cos^{4}x}\, dx$ .

Problem 2038

Find the integral: $\displaystyle \int \dfrac {\sin x\cos x}{1+\sin^{4}x}\, dx$ .

Problem 2039

Find the integral: $\displaystyle \int \dfrac {dx}{\sin^{6}x+\cos^{6}x}$ .

Problem 2040

Find the integral: $\displaystyle \int \dfrac {dx}{(\sin^{2}x+2\cos^{2}x)^{2}}$ .

Problem 2041

Find the integral: $\displaystyle \int \dfrac {dx}{a\sin x+b\cos x}$ (first reduce the denominator to the form of a logarithm).

Problem 2042

Prove that $\displaystyle \int \dfrac {a_{1}\sin x+b_{1}\cos x}{a\sin x+b\cos x}\, dx=Ax+B\ln \left|a\sin x+b\cos x\right|+C$ , where $A$ , $B$ , $C$ are constants.

Problem 2043

Find the integral: $\displaystyle \int \dfrac {\sin x-\cos x}{\sin x+2\cos x}\, dx$ .

Problem 2043.1

[NO TASK]

Problem 2043.1

[NO TASK]

Problem 2044

Find the integral: $\displaystyle \int \dfrac {dx}{3+5\operatorname {tg}x}$ .

Problem 2045

Find the integral: $\displaystyle \int \dfrac {a_{1}\sin x+b_{1}\cos x}{(a\sin x+b\cos x)^{2}}\, dx$ .

Problem 2046

Prove that: $\displaystyle \int \dfrac {a_{1}\sin x+b_{1}\cos x+c_{1}}{a\sin x+b\cos x+c}\, dx=Ax+B\ln \left|a\sin x+b\cos x+c\right|+C\displaystyle \int \dfrac {dx}{a\sin x+b\cos x+c}$ , where $A$ , $B$ , $C$ are all constant coefficients.

Problem 2047

Find the integral: $\displaystyle \int \dfrac {\sin x+2\cos x-3}{\sin x-2\cos x+3}\, dx$ .

Problem 2048

Find the integral: $\displaystyle \int \dfrac {\sin x\, dx}{\sqrt{2}+\sin x+\cos x}$ .

Problem 2049

Find the integral: $\displaystyle \int \dfrac {2\sin x+\cos x}{3\sin x+4\cos x-2}\, dx$ .

Problem 2050

Prove that: $\displaystyle \int \dfrac {a_{1}\sin^{2}x+2b_{1}\sin x\cos x+c_{1}\cos^{2}x}{a\sin x+b\cos x}\, dx=A\sin x+B\cos x+C\displaystyle \int \dfrac {dx}{a\sin x+b\cos x}$ , where $A$ , $B$ , $C$ are all constant coefficients.

Problem 2051

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x-4\sin x\cos x+3\cos^{2}x}{\sin x+\cos x}\, dx$ .

Problem 2052

Find the integral: $\displaystyle \int \dfrac {\sin^{2}x-\sin x\cos x+2\cos^{2}x}{\sin x+2\cos x}\, dx$ .

Problem 2053

Prove that if $(a-c)^{2}+b^{2}\neq 0$ , then

\int \dfrac {a_{1}\sin x+b_{1}\cos x}{a\sin ^{2}x+2b\sin x\cos x+c\cos ^{2}x}\, dx=A\int \dfrac {du_{1}}{k_{1}u_{1}^{2}+\lambda _{1}}+B\int \dfrac {du_{2}}{k_{2}u_{2}^{2}+\lambda _{2}} ,

where $A$ , $B$ are undetermined coefficients, $\lambda_{1}$ , $\lambda_{2}$ are the roots of the equation below

\begin{vmatrix} a-\lambda & b \\ b & c-\lambda \end{vmatrix}=0\quad (\lambda _{1}\neq \lambda _{2})

and

u_{i}=(a-\lambda _{i})\sin x+b\cos x ，\quad k_{i}=\dfrac {1}{a-\lambda _{i}}\quad (i=1,2) .

Problem 2054

Find the integral: $\displaystyle \int \dfrac {2\sin x-\cos x}{3\sin^{2}x+4\cos^{2}x}\, dx$ .

Problem 2055

Find the integral: $\displaystyle \int \dfrac {\sin x+\cos x}{2\sin^{2}x-4\sin x\cos x+5\cos^{2}x}\, dx$ .

Problem 2056

Evaluate the integral: $\displaystyle \int \dfrac {\sin x-2\cos x}{1+4\sin x\cos x}\, dx$ .

Problem 2057

Prove that

\int \dfrac {dx}{(a\sin x+b\cos x)^{n}}=\dfrac {A\sin x+B\cos x}{(a\sin x+b\cos x)^{n-1}}+C\int \dfrac {dx}{(a\sin x+b\cos x)^{n-2}} ,

where $A$ , $B$ , $C$ are undetermined coefficients.

Problem 2058

Evaluate $\displaystyle \int \dfrac {dx}{(\sin x+2\cos x)^{3}}$ .

Problem 2059

If $n$ is a natural number greater than $1$ , prove that

\int \dfrac {dx}{(a+b\cos x)^{n}}=\dfrac {A\sin x}{(a+b\cos x)^{n-1}}+B\int \dfrac {dx}{(a+b\cos x)^{n-1}}+C\int \dfrac {dx}{(a+b\cos x)^{n-2}}\quad (\left|a\right|\neq \left|b\right|) ,

and find the coefficients $A$ , $B$ and $C$ .

Problem 2060

Evaluate the integral: $\displaystyle \int \dfrac {\sin x\, dx}{\cos x\sqrt{1+\sin^{2}x}}$ .

Problem 2061

Evaluate the integral: $\displaystyle \int \dfrac {\sin^{2}x}{\cos^{2}x\sqrt{\operatorname {tg}x}}\, dx$ .

Problem 2062

Evaluate the integral: $\displaystyle \int \dfrac {\sin x\, dx}{\sqrt{2+\sin 2x}}$ .

Problem 2063

Evaluate the integral: $\displaystyle \int \dfrac {dx}{(1+\varepsilon \cos x)^{2}}$ ( $0<\varepsilon <1$ ).

Problem 2064

Evaluate the integral: $\displaystyle \int \dfrac {\cos^{n-1}\dfrac {x+a}{2}}{\sin^{n+1}\dfrac {x-a}{2}}\, dx$ .

Problem 2065

Derive a recurrence formula for the integral $\displaystyle I_{n}=\int \left(\dfrac {\sin \dfrac {x-a}{2}}{\sin \dfrac {x+a}{2}}\right)^{n}\, dx$ ( $n$ a natural number).

Section 3.5Integration of various transcendental functions[63/95%]

Problem 2066

Prove that if $P(x)$ is a polynomial of degree $n$ , then

\int P(x)e^{ax}\, dx=e^{ax}\left[\dfrac {P(x)}{a}-\dfrac {P’(x)}{a^{2}}+\cdots +(-1)^{n}\dfrac {P^{(n)}(x)}{a^{n+1}}\right]+C .

Problem 2067

Prove that if $P(x)$ is a polynomial of degree $n$ , then

\begin{align} \int P(x)\cos ax\, dx & =\dfrac {\sin ax}{a}\left[P(x)-\dfrac {P”(x)}{a^{2}}+\dfrac {P^{(4)}(x)}{a^{4}}-\cdots \right] \\ & \quad +\dfrac {\cos ax}{a^{2}}\left[P'(x)-\dfrac {P”’(x)}{a^{2}}+\dfrac {P^{(5)}(x)}{a^{4}}-\cdots \right]+C \end{align}

and

\begin{align} \int P(x)\sin ax\, dx & =-\dfrac {\cos ax}{a}\left[P(x)-\dfrac {P”(x)}{a^{2}}+\dfrac {P^{(4)}(x)}{a^{4}}-\cdots \right] \\ & \quad +\dfrac {\sin ax}{a^{2}}\left[P'(x)-\dfrac {P”’(x)}{a^{2}}+\dfrac {P^{(5)}(x)}{a^{4}}-\cdots \right]+C . \end{align}

Problem 2068

Evaluate the integral: $\displaystyle \int x^{3}e^{3x}\, dx$ .

Problem 2069

Evaluate the integral: $\displaystyle \int (x^{2}-2x+2)e^{-x}\, dx$ .

Problem 2070

Evaluate the integral: $\displaystyle \int x^{5}\sin 5x\, dx$ .

Problem 2071

Evaluate the integral: $\displaystyle \int (1+x^{2})^{2}\cos x\, dx$ .

Problem 2072

Evaluate the integral: $\displaystyle \int x^{7}e^{-x^{2}}\, dx$ .

Problem 2073

Evaluate the integral: $\displaystyle \int x^{2}e^{\sqrt{x}}\, dx$ .

Problem 2074

Evaluate the integral: $\displaystyle \int e^{ax}\cos^{2}bx\, dx$ .

Problem 2075

Evaluate the integral: $\displaystyle \int e^{ax}\sin^{3}bx\, dx$ .

Problem 2076

Evaluate the integral: $\displaystyle \int xe^{x}\sin x\, dx$ .

Problem 2077

Evaluate the integral: $\displaystyle \int x^{2}e^{x}\cos x\, dx$ .

Problem 2078

Evaluate the integral: $\displaystyle \int xe^{x}\sin^{2}x\, dx$ .

Problem 2079

Evaluate the integral: $\displaystyle \int (x-\sin x)^{3}\, dx$ .

Problem 2080

Evaluate the integral: $\displaystyle \int \cos^{2}\sqrt{x}\, dx$ .

Problem 2081

Prove that if $R$ is a rational function and $a_{1},a_{2},\ldots ,a_{n}$ are commensurable numbers, then the integral

\int R(e^{a_{1}x},e^{a_{2}x},\ldots ,e^{a_{n}x})\, dx

is an elementary function.

Problem 2082

Evaluate the following integral: $\displaystyle \int \dfrac {dx}{(1+e^{x})^{2}}$ .

Problem 2083

Evaluate the following integral: $\displaystyle \int \dfrac {e^{2x}\, dx}{1+e^{x}}$ .

Problem 2084

Evaluate the following integral: $\displaystyle \int \dfrac {dx}{e^{2x}+e^{x}-2}$ .

Problem 2085

Evaluate the following integral: $\displaystyle \int \dfrac {dx}{1+e^{\frac{x}{2}}+e^{\frac{x}{3}}+e^{\frac{x}{6}}}$ .

Problem 2086

Evaluate the following integral: $\displaystyle \int \dfrac {1+e^{\frac{x}{2}}}{(1+e^{\frac{x}{4}})^{2}}\, dx$ .

Problem 2087

Evaluate the following integral: $\displaystyle \int \dfrac {dx}{\sqrt{e^{x}-1}}$ .

Problem 2088

Evaluate the following integral: $\displaystyle \int \sqrt{\dfrac {e^{x}-1}{e^{x}+1}}\, dx$ .

Problem 2089

Evaluate the following integral: $\displaystyle \int \sqrt{e^{2x}+4e^{x}-1}\, dx$ .

Problem 2090

Evaluate the following integral: $\displaystyle \int \dfrac {dx}{\sqrt{1+e^{x}}+\sqrt{1-e^{x}}}$ .

Problem 2091

Prove that the integral

\int R(x)e^{ax}\, dx ,

(where $R$ is a rational function whose denominator has only real roots) can be expressed by elementary functions together with the transcendental function

\int \dfrac {e^{ax}}{x}\, dx=\operatorname {li}(e^{ax})+C ,\quad \text{where }\operatorname {li}x=\int \dfrac {dx}{\ln x}

Problem 2092

Under what circumstances is the integral

\int P\left(\dfrac {1}{x}\right)e^{x}\, dx

(where $P\left(\dfrac {1}{x}\right)=a_{0}+\dfrac {a_{1}}{x}+\cdots +\dfrac {a_{n}}{x^{n}}$ and $a_{0},a_{1},\ldots ,a_{n}$ are constants) an elementary function?

Problem 2093

Evaluate the integral: $\displaystyle \int \left(1-\dfrac {2}{x}\right)^{2}e^{x}\, dx$ .

Problem 2094

Evaluate the integral: $\displaystyle \int \left(1-\dfrac {1}{x}\right)e^{-x}\, dx$ .

Problem 2095

Evaluate the integral: $\displaystyle \int \dfrac {e^{2x}}{x^{2}-3x+2}\, dx$ .

Problem 2096

Evaluate the integral: $\displaystyle \int \dfrac {xe^{x}}{(x+1)^{2}}\, dx$ .

Problem 2097

Evaluate the integral: $\displaystyle \int \dfrac {x^{4}e^{2x}}{(x-2)^{2}}\, dx$ .

Problem 2098

Find integrals involving $\ln f(x),\operatorname {arctg}f(x),\arcsin f(x),\arccos f(x)$ and similar functions, where $f(x)$ is an algebraic function: $\displaystyle \int \ln^{n}x\, dx$ ( $n$ a natural number).

Problem 2099

Find integrals involving $\ln f(x),\operatorname {arctg}f(x),\arcsin f(x),\arccos f(x)$ and similar functions, where $f(x)$ is an algebraic function: $\displaystyle \int x^{3}\ln^{3}x\, dx$ .

Problem 2100

Problem 2101

Evaluate the integral: $\displaystyle \int \ln \left[(x+a)^{x+a}(x+b)^{x+b}\right]\cdot \dfrac {dx}{(x+a)(x+b)}$ .

Problem 2102

Evaluate the integral: $\displaystyle \int \ln^{2}(x+\sqrt{1+x^{2}})\, dx$ .

Problem 2103

Evaluate the integral: $\displaystyle \int \ln (\sqrt{1-x}+\sqrt{1+x})\, dx$ .

Problem 2104

Evaluate the integral: $\displaystyle \int \dfrac {\ln x}{(1+x^{2})^{\frac{3}{2}}}\, dx$ .

Problem 2105

Evaluate the integral: $\displaystyle \int x\operatorname {arctg}(x+1)\, dx$ .

Problem 2106

Evaluate the integral: $\displaystyle \int \sqrt{x}\operatorname {arctg}\sqrt{x}\, dx$ .

Problem 2107

Evaluate the integral: $\displaystyle \int x\arcsin (1-x)\, dx$ .

Problem 2108

Evaluate the integral: $\displaystyle \int \arcsin \sqrt{x}\, dx$ .

Problem 2109

Evaluate the integral: $\displaystyle \int x\arccos \dfrac {1}{x}\, dx$ .

Problem 2110

Evaluate the integral: $\displaystyle \int \arcsin \dfrac {2\sqrt{x}}{1+x}\, dx$ .

Problem 2111

Evaluate the integral: $\displaystyle \int \dfrac {\arccos x}{(1-x^{2})^{\frac{3}{2}}}\, dx$ .

Problem 2112

Evaluate the integral: $\displaystyle \int \dfrac {x\arccos x}{(1-x^{2})^{\frac{3}{2}}}\, dx$ .

Problem 2113

Evaluate the integral: $\displaystyle \int x\operatorname {arctg}x\ln (1+x^{2})\, dx$ .

Problem 2114

Evaluate the integral: $\displaystyle \int x\ln \dfrac {1+x}{1-x}\, dx$ .

Problem 2115

Evaluate the integral: $\displaystyle \int \dfrac {\ln (x+\sqrt{1+x^{2}})}{(1+x^{2})^{\frac{3}{2}}}\, dx$ .

Problem 2116

Find integrals involving hyperbolic functions: $\displaystyle \int \operatorname {sh}^{2}x\operatorname {ch}^{2}x\, dx$ .

Problem 2117

Find integrals involving hyperbolic functions: $\displaystyle \int \operatorname {ch}^{4}x\, dx$ .

Problem 2118

Find integrals involving hyperbolic functions: $\displaystyle \int \operatorname {sh}^{3}x\, dx$ .

Problem 2119

Find integrals involving hyperbolic functions: $\displaystyle \int \operatorname {sh}x\operatorname {sh}2x\operatorname {sh}3x\, dx$ .

Problem 2120

Find integrals involving hyperbolic functions: $\displaystyle \int \operatorname {th}x\, dx$ .

Problem 2121

Find integrals involving hyperbolic functions: $\displaystyle \int \operatorname {cth}^{2}x\, dx$ .

Problem 2122

Find integrals involving hyperbolic functions: $\displaystyle \int \sqrt{\operatorname {th}x}\, dx$ .

Problem 2123

Find integrals involving hyperbolic functions: $\displaystyle \int \dfrac {dx}{\operatorname {sh}x+2\operatorname {ch}x}$ .

Problem 2123.1

[NO TASK]

Problem 2123.1

[NO TASK]

Problem 2123.2

[NO TASK]

Problem 2123.2

[NO TASK]

Problem 2123.3

[NO TASK]

Problem 2123.3

[NO TASK]

Problem 2124

Find integrals involving hyperbolic functions: $\displaystyle \int \operatorname {sh}ax\sin bx\, dx$ .

Problem 2125

Find integrals involving hyperbolic functions: $\displaystyle \int \operatorname {sh}ax\cos bx\, dx$ .

Section 3.6Various examples of the integration of functions[56/98%]

Problem 2126

Evaluate the integral: $\displaystyle \int \dfrac {dx}{x^{6}(1+x^{2})}$ .

Problem 2127

Evaluate the integral: $\displaystyle \int \dfrac {x^{2}\, dx}{(1-x^{2})^{3}}$ .

Problem 2128

Evaluate the integral: $\displaystyle \int \dfrac {dx}{1+x^{4}+x^{8}}$ .

Problem 2129

Evaluate the integral: $\displaystyle \int \dfrac {dx}{\sqrt{x}+\sqrt[3]{x}}$ .

Problem 2130

Evaluate the integral: $\displaystyle \int x^{2}\sqrt{\dfrac {x}{1-x}}\, dx$ .

Problem 2131

Evaluate the integral: $\displaystyle \int \dfrac {x+2}{x^{2}\sqrt{1-x^{2}}}\, dx$ .

Problem 2132

Find the integral: $\displaystyle \int \sqrt{\dfrac {x}{1-x\sqrt{x}}}\, dx$ .

Problem 2133

Find the integral: $\displaystyle \int \dfrac {x^{5}\, dx}{\sqrt{1+x^{2}}}$ .

Problem 2134

Find the integral: $\displaystyle \int \dfrac {dx}{\sqrt[3]{x^{2}(1-x)}}$ .

Problem 2135

Find the integral: $\displaystyle \int \dfrac {dx}{x\sqrt{1+x^{3}+x^{6}}}$ .

Problem 2136

Find the integral: $\displaystyle \int \dfrac {dx}{x\sqrt{x^{4}-2x^{2}-1}}$ .

Problem 2137

Find the integral: $\displaystyle \int \dfrac {1+\sqrt{1-x^{2}}}{1-\sqrt{1-x^{2}}}\, dx$ .

Problem 2138

Find the integral: $\displaystyle \int \dfrac {(1+x)\, dx}{x+\sqrt{x+x^{2}}}$ .

Problem 2139

Find the integral: $\displaystyle \int \dfrac {\ln (1+x+x^{2})}{(1+x)^{2}}\, dx$ .

Problem 2140

Find the integral: $\displaystyle \int (2x+3)\arccos (2x-3)\, dx$ .

Problem 2141

Find the integral: $\displaystyle \int x\ln (4+x^{4})\, dx$ .

Problem 2142

Find the integral: $\displaystyle \int \dfrac {\arcsin x}{x^{2}}\cdot \dfrac {1+x^{2}}{\sqrt{1-x^{2}}}\, dx$ .

Problem 2143

Find the integral: $\displaystyle \int \dfrac {x\ln (1+\sqrt{1+x^{2}})}{\sqrt{1+x^{2}}}\, dx$ .

Problem 2144

Find the integral: $\displaystyle \int x\sqrt{x^{2}+1}\ln \sqrt{x^{2}-1}\, dx$ .

Problem 2145

Find the integral: $\displaystyle \int \dfrac {x}{\sqrt{1-x^{2}}}\ln \dfrac {x}{\sqrt{1-x}}\, dx$ .

Problem 2146

Find the integral: $\displaystyle \int \dfrac {dx}{(2+\sin x)^{2}}$ .

Problem 2147

Find the integral: $\displaystyle \int \dfrac {\sin 4x}{\sin^{8}x+\cos^{8}x}\, dx$ .

Problem 2148

Find the integral: $\displaystyle \int \dfrac {dx}{\sin x\sqrt{1+\cos x}}$ .

Problem 2149

Find the integral: $\displaystyle \int \dfrac {ax^{2}+b}{x^{2}+1}\operatorname {arctg}x\, dx$ .

Problem 2150

Find the integral: $\displaystyle \int \dfrac {ax^{2}+b}{x^{2}-1}\ln \left|\dfrac {x-1}{x+1}\right|\, dx$ .

Problem 2151

Find the integral: $\displaystyle \int \dfrac {x\ln x}{(1+x^{2})^{2}}\, dx$ .

Problem 2152

Find the integral: $\displaystyle \int \dfrac {x\operatorname {arctg}x}{\sqrt{1+x^{2}}}\, dx$ .

Problem 2153

Find the integral: $\displaystyle \int \dfrac {\sin 2x\, dx}{\sqrt{1+\cos^{4}x}}$ .

Problem 2154

Find the integral: $\displaystyle \int \dfrac {x^{3}\arccos x}{\sqrt{1-x^{2}}}\, dx$ .

Problem 2155

Find the integral: $\displaystyle \int \dfrac {x^{4}\operatorname {arctg}x}{1+x^{2}}\, dx$ .

Problem 2156

Find the integral: $\displaystyle \int \dfrac {x\operatorname {arcctg}x}{(1+x^{2})^{2}}\, dx$ .

Problem 2157

Find the integral: $\displaystyle \int \dfrac {x\ln (x+\sqrt{1+x^{2}})}{(1-x^{2})^{2}}\, dx$ .

Problem 2158

Find the integral: $\displaystyle \int \sqrt{1-x^{2}}\arcsin x\, dx$ .

Problem 2159

Find the integral: $\displaystyle \int x(1+x^{2})\operatorname {arcctg}x\, dx$ .

Problem 2160

Find the integral: $\displaystyle \int x^{x}(1+\ln x)\, dx$ .

Problem 2161

Find the integral: $\displaystyle \int \dfrac {\arcsin e^{x}}{e^{x}}\, dx$ .

Problem 2162

Find the integral: $\displaystyle \int \dfrac {\operatorname {arctg}e^{\frac{x}{2}}}{e^{\frac{x}{2}}(1+e^{x})}\, dx$ .

Problem 2163

Find the integral: $\displaystyle \int \dfrac {dx}{(e^{x+1}+1)^{2}-(e^{x-1}+1)^{2}}$ .

Problem 2164

Find the integral: $\displaystyle \int \sqrt{\operatorname {th}^{2}x+1}\, dx$ .

Problem 2165

Find the integral: $\displaystyle \int \dfrac {1+\sin x}{1+\cos x}\cdot e^{x}\, dx$ .

Problem 2166

Find the integral: $\displaystyle \int \left|x\right|\, dx$ .

Problem 2167

Find the integral: $\displaystyle \int x\left|x\right|\, dx$ .

Problem 2168

Find the integral: $\displaystyle \int (x+\left|x\right|)^{2}\, dx$ .

Problem 2169

Find the integral: $\displaystyle \int (\left|1+x\right|-\left|1-x\right|)\, dx$ .

Problem 2170

Find the integral: $\displaystyle \int e^{-\left|x\right|}\, dx$ .

Problem 2171

Find the integral: $\displaystyle \int \max (1,x^{2})\, dx$ .

Problem 2172

Find the integral: $\displaystyle \int \varphi (x)\, dx$ , where $\varphi (x)$ is the distance from the number $x$ to its nearest integer.

Problem 2173

Find the integral: $\displaystyle \int [x]\left|\sin \pi x\right|\, dx \quad (x\geqslant 0)$ .

Problem 2174

Find the integral: $\displaystyle \int f(x)\, dx$ , where

f(x)= \begin{cases} 1-x^{2}, & \text{when }\left|x\right|\leqslant 1 , \\ 1-\left|x\right|, & \text{when }\left|x\right|>1 . \end{cases}

Problem 2175

Find the integral: $\displaystyle \int f(x)\, dx$ , where

f(x)= \begin{cases} 1, & \text{if }-\infty <x\leqslant 0 , \\ x+1, & \text{if }0\leqslant x\leqslant 1 , \\ 2x, & \text{if }1<x<+\infty . \end{cases}

Problem 2176

Find the integral: $\displaystyle \int xf''(x)\, dx$ .

Problem 2177

Find the integral: $\displaystyle \int f'(2x)\, dx$ .

Problem 2178

Suppose $f'(x^{2})=\dfrac {1}{x}$ ( $x>0$ ), find $f(x)$ .

Problem 2179

Suppose $f'(\sin^{2}x)=\cos^{2}x$ , find $f(x)$ .

Problem 2180

Suppose

f'(\ln x)= \begin{cases} 1, & \text{when }0<x\leqslant 1 , \\ x, & \text{when }1<x<+\infty \end{cases}

and $f(0)=0$ , find $f(x)$ .

Problem 2180.1

(a)(b)(c)(d)

(a)

(b)

(c)

(d)

Problem 2180.1

(a)(b)(c)(d)

(a)

(b)

(c)

(d)