Math Problems Vault

Problem 1628

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

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Problem 1629

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

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Problem 1630

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

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Problem 1631

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

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Problem 1632

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

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Problem 1633

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

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Problem 1634

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

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Problem 1635

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

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Problem 1636

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

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Problem 1637

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

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Problem 1638

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

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Problem 1639

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

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Problem 1640

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

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Problem 1641

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

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Problem 1642

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

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Problem 1643

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

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Problem 1644

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

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Problem 1645

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

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Problem 1646

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

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Problem 1647

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

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Problem 1648

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

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Problem 1649

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

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Problem 1650

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

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Problem 1651

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

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Problem 1652

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

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Problem 1653

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

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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) .

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Problem 1655

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

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Problem 1656

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

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Problem 1657

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

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Problem 1658

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

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Problem 1659

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

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Problem 1660

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

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Problem 1661

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

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Problem 1662

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

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Problem 1663

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

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Problem 1664

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

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Problem 1665

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

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Problem 1666

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

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Problem 1667

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

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Problem 1668

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

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Problem 1669

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

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Problem 1670

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

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Problem 1671

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

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Problem 1672

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

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Problem 1673

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

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Problem 1674

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

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Problem 1675

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

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Problem 1676

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

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Problem 1677

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

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Problem 1678

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

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Problem 1679

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

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Problem 1680

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

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Problem 1681

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

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Problem 1682

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

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Problem 1683

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

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Problem 1684

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

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Problem 1685

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

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Problem 1686

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

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Problem 1687

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

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Problem 1688

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

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Problem 1689

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

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Problem 1690

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

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Problem 1691

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

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Problem 1692

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

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Problem 1693

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

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Problem 1694

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

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Problem 1695

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

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Problem 1696

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

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Problem 1697

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

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Problem 1698

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

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Problem 1699

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

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Problem 1700

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

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Problem 1700.1

[NO TASK]

Problem 1700.2

[NO TASK]

Problem 1700.3

[NO TASK]

Problem 1701

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

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Problem 1702

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

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Problem 1703

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

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Problem 1704

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

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Problem 1705

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

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Problem 1706

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

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Problem 1707

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

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Problem 1708

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

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Problem 1709

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

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Problem 1710

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

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Problem 1711

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

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Problem 1712

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

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Problem 1713

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

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Problem 1714

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

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Problem 1715

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

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Problem 1716

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

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Problem 1717

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

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Problem 1718

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

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Problem 1719

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

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Problem 1720

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

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Problem 1721

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

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Problem 1721.1

[NO TASK]

Problem 1722

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

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Problem 1723

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

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Problem 1724

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

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Problem 1725

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

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Problem 1726

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

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Problem 1727

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

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Problem 1728

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

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Problem 1729

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

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Problem 1730

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

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Problem 1731

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

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Problem 1732

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

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Problem 1733

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

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Problem 1734

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

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Problem 1735

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

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Problem 1736

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

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Problem 1737

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

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Problem 1738

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

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Problem 1739

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

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Problem 1740

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

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Problem 1741

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

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Problem 1742

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

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Problem 1743

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

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Problem 1744

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

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Problem 1745

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

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Problem 1746

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

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Problem 1747

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

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Problem 1748

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

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Problem 1749

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

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Problem 1750

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

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Problem 1751

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

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Problem 1752

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

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Problem 1753

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

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Problem 1754

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

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Problem 1755

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

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Problem 1756

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

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Problem 1757

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

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Problem 1758

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

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Problem 1759

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

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Problem 1760

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

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Problem 1761

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

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Problem 1762

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

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Problem 1763

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

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Problem 1764

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

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Problem 1765

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

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Problem 1766

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

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Problem 1767

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

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Problem 1768

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

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Problem 1769

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

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Problem 1770

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

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Problem 1771

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

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Problem 1772

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

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Problem 1773

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

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Problem 1774

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

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Problem 1775

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

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Problem 1776

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

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Problem 1777

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

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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: $\int \dfrac {dx}{(1-x^{2})^{\frac{3}{2}}}$ .

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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: $\int \dfrac {x^{2} dx}{\sqrt{x^{2}-2}}$ .

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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: $\int \sqrt{a^{2}-x^{2}}\, dx$ .

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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: $\int \dfrac {dx}{(x^{2}+a^{2})^{\frac{3}{2}}}$ .

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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: $\int \sqrt{\dfrac {a+x}{a-x}}\, dx$ .

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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: $\int x\sqrt{\dfrac {x}{2a-x}}\, dx$ .

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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: $\int \dfrac {dx}{\sqrt{(x-a)(b-x)}}$ .

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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: $\int \sqrt{(x-a)(b-x)}\, dx$ .

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Problem 1786

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

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Problem 1787

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

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Problem 1788

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

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Problem 1789

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

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Problem 1790

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

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Problem 1791

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

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Problem 1792

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

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Problem 1793

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

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Problem 1794

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

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Problem 1795

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

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Problem 1796

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

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Problem 1797

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

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Problem 1798

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

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Problem 1799

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

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Problem 1800

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

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Problem 1801

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

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Problem 1802

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

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Problem 1803

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

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Problem 1804

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

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Problem 1805

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

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Problem 1806

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

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Problem 1807

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

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Problem 1808

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

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Problem 1809

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

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Problem 1810

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

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Problem 1811

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

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Problem 1812

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

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Problem 1813

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

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Problem 1814

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

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Problem 1815

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

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Problem 1816

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

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Problem 1817

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

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Problem 1818

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

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Problem 1819

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

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Problem 1820

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

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Problem 1821

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

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Problem 1822

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

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Problem 1823

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

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Problem 1824

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

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Problem 1825

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

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Problem 1826

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

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Problem 1827

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

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Problem 1828

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

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Problem 1829

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

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Problem 1830

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

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Problem 1831

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

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Problem 1832

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

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Problem 1833

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

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Problem 1834

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

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Problem 1835

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

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

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Problem 1837

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

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Problem 1838

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

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Problem 1839

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

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Problem 1840

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

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Problem 1841

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

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Problem 1842

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

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Problem 1843

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

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Problem 1844

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

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Problem 1845

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

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Problem 1846

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

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Problem 1847

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

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Problem 1848

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

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Problem 1849

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

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

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Problem 1851

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

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Problem 1852

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

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Problem 1853

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

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Problem 1853.1

[NO TASK]

Problem 1854

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

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Problem 1855

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

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Problem 1856

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

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Problem 1857

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

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Problem 1858

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

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Problem 1859

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

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Problem 1860

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

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Problem 1861

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

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Problem 1862

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

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Problem 1863

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

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Problem 1864

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

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Problem 1865

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

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