Chapter 7

Integration of Functions

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Section 7.1Basic Formulas and Techniques of Integration[102/100%]

Problem 2797

Applying formulas 1 and 2 and rules I and II, find the integral $\int x^{3} dx$ .

Problem 2798

Applying formulas 1 and 2 and rules I and II, find the integral $\int \sqrt{x}\, dx$ .

Problem 2799

Applying formulas 1 and 2 and rules I and II, find the integral $\int x \sqrt{x}\, dx$ .

Problem 2800

Applying formulas 1 and 2 and rules I and II, find the integral $\int \sqrt{x \sqrt{x \sqrt{x}}}\, dx$ .

Problem 2801

Find the integral $\int \frac{dx}{\sqrt{x}}$ .

Problem 2802

Find the integral $\int \frac{dx}{\sqrt[3]{x}}$ .

Problem 2803

Find the integral $\int \frac{dx}{x^{3}}$ .

Problem 2804

Find the integral $\int \frac{dx}{x \sqrt{x}}$ .

Problem 2805

Find the integral $\int (x^{2}+6x-5)\, dx$ .

Problem 2806

Find the integral $\int (x^{4}-3x^{2}+5x)\, dx$ .

Problem 2807

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

Problem 2808

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

Problem 2809

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

Problem 2810

Find the integral $\int \frac{x^{3}+2x^{2}-x+3}{x}\, dx$ .

Problem 2811

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

Problem 2812

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

Problem 2813

The following integrals are found directly by applying formulas 1--12, and also by means of the change of variable $ax+b=t$ . Find the integral $\int e^{-x}\, dx$ .

Problem 2814

Find the integral $\int e^{-3x}\, dx$ .

Problem 2815

Find the integral $\int \sin 2x\, dx$ .

Problem 2816

Find the integral $\int \cos (3x-5)\, dx$ .

Problem 2817

Find the integral $\int \frac{dx}{x-2}$ .

Problem 2818

Find the integral $\int \frac{dx}{3x+2}$ .

Problem 2819

Find the integral $\int \frac{x^{2}-x+1}{x-2}\, dx$ .

Problem 2820

Find the integral $\int \frac{dx}{x^{2}+2}$ .

Problem 2821

Find the integral $\int \frac{dx}{x^{3}-3}$ .

Problem 2822

Find the integral $\int \frac{dx}{2x^{2}+7}$ .

Problem 2823

Find the integral $\int \frac{dx}{\sqrt{5-x^{2}}}$ .

Problem 2824

Find the integral $\int \frac{x^{4}\, dx}{x-1}$ .

Problem 2825

Find the integral $\int \frac{dx}{\sqrt{x^{2}-6}}$ .

Problem 2826

Find the integral $\int \frac{dx}{\sqrt{x^{2}+7}}$ .

Problem 2827

Find the integral $\int \frac{dx}{(x-2)^{3}}$ .

Problem 2828

Find the integral $\int \frac{dx}{(2x-1)^{2}}$ .

Problem 2829

Find the integral $\int \frac{dx}{\cos^{2} 3x}$ .

Problem 2830

Find the integral $\int \frac{dx}{1+\cos x}$ .

Problem 2831

Find the integral $\int \frac{dx}{\sin^{2}(5x-2)}$ .

Problem 2832

Find the integral $\int \frac{dx}{\sqrt{2-3x}}$ .

Problem 2833

Find the integral $\int \frac{dx}{\sqrt{2-3x^{2}}}$ .

Problem 2834

Find the integral $\int \frac{dx}{\cos x}$ .

Problem 2835

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

Problem 2836

Find the integral $\int \frac{dx}{\sqrt{x-1}-\sqrt{x-2}}$ , by rationalizing the numerator.

Problem 2837

Find the integral $\int \frac{dx}{(x+2)^{2}+3}$ .

Problem 2838

Find the integral $\int \frac{dx}{x^{2}+6x+13}$ .

Problem 2839

Find the integral $\int \frac{dx}{x^{2}+5x+6}$ .

Problem 2840

Find the integral $\int \frac{dx}{\sqrt{x^{2}+2x+3}}$ .

Problem 2841

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

Problem 2842

Find the integral $\int \frac{dx}{\sqrt{2x^{2}+3x}}$ .

Problem 2843

Find the integral $\int \frac{dx}{\sqrt{2ax-x^{2}}}$ .

Problem 2844

In a number of cases it is convenient to transform products and powers of sines and cosines into sums or differences. Find the integral $\int \cos x \cos 3x\, dx$ .

Problem 2845

Find the integral $\int \cos x \sin 3x\, dx$ .

Problem 2846

Find the integral $\int \cos 3x \cos 4x\, dx$ .

Problem 2847

Find the integral $\int \cos x \cos 3x \cos 5x\, dx$ .

Problem 2848

Find the integral $\int \sin x \sin 2x \sin 3x\, dx$ .

Problem 2849

Find the integral $\int \sin \frac{x}{2}\cos \frac{x}{12}\, dx$ .

Problem 2850

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

Problem 2851

Find the integral $\int \sin^{4} x\, dx$ .

Problem 2852

Find the integral $\int \cos^{4} x\, dx$ .

Problem 2853

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

Problem 2854

The following integrals are found by simple substitutions. Find the integral $\int \frac{e^{x}\, dx}{\sqrt{e^{x}+1}}$ .

Problem 2855

Find the integral $\int \frac{e^{x}\, dx}{e^{2x}+1}$ .

Problem 2856

Find the integral $\int \frac{\ln x}{x}\, dx$ .

Problem 2857

Find the integral $\int \frac{dx}{x \ln x}$ .

Problem 2858

Find the integral $\int \frac{\arctg x}{1+x^{2}}\, dx$ .

Problem 2859

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

Problem 2860

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

Problem 2861

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

Problem 2862

Find the integral $\int \frac{x\, dx}{(x^{2}+1)^{2}}$ .

Problem 2863

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

Problem 2864

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

Problem 2865

Find the integral $\int \frac{\cos x\, dx}{\sqrt[3]{\sin^{2}x}}$ .

Problem 2866

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

Problem 2867

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

Problem 2868

Find the integral $\int \frac{\sin 2x\, dx}{\sin^{2}x+3}$ .

Problem 2869

Find the integral $\int \frac{\sqrt{\tg x}\, dx}{\cos^{2}x}$ .

Problem 2870

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

Problem 2871

Find the integral $\int \frac{dx}{\sin^{2}x+5\cos^{2}x}$ .

Problem 2872

Find the integral $\int \tg x\, dx$ .

Problem 2873

Find the integral $\int \tg^{2}x\, dx$ .

Problem 2874

Find the integral $\int \frac{2x+11}{x^{2}+6x+13}\, dx$ .

Problem 2875

Find the integral $\int \frac{x\, dx}{x^{2}+x+1}$ .

Problem 2876

Find the integral $\int \frac{4x+8}{3x^{2}+2x+5}\, dx$ .

Problem 2877

Find the integral $\int \frac{5x+7}{\sqrt{5-4x-x^{2}}}\, dx$ .

Problem 2878

Find the integral $\int \frac{3x+2}{\sqrt{3+x+x^{2}}}\, dx$ .

Problem 2879

Find the integral $\int \frac{x\, dx}{\sqrt{x^{2}-ax}}$ .

Problem 2880

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

Problem 2881

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

Problem 2882

Find the integral $\int \frac{(x+a)\, dx}{\sqrt{ax-x^{2}}}$ .

Problem 2883

Using one of the trigonometric substitutions $x=a\sin \varphi$ , $x=a\tg \varphi$ , $x=a\sin^{2}\varphi$ , find the integral $\int \frac{dx}{(a^{2}+x^{2})\sqrt{a^{2}+x^{2}}}$ .

Problem 2884

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

Problem 2885

Find the integral $\int \frac{dx}{\sqrt{x(a-x)}}$ .

Problem 2886

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

Problem 2887

Find the integral $\int \sqrt{\frac{1-x}{1+x}}\, dx$ , by rationalizing the denominator.

Problem 2888

Applying integration by parts, expressed by equality III, find the integral $\int x^{2}\ln x\, dx$ .

Problem 2889

Find the integral $\int x\cos x\, dx$ .

Problem 2890

Find the integral $\int x\ch x\, dx$ .

Problem 2891

Find the integral $\int x\sin x\, dx$ .

Problem 2892

Find the integral $\int x\arctg x\, dx$ .

Problem 2893

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

Problem 2894

Find the integral $\int x^{2}e^{-x}\, dx$ .

Problem 2895

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

Problem 2896

Find the integral $\int x\ln (x^{2}-1)\, dx$ .

Problem 2897

Find the integral $\int \ln x\, dx$ .

Problem 2898

Find the integral $\int x\ln^{2}x\, dx$ .

Section 7.2Integration of Rational Functions[76/97%]

Problem 2899

Find the integral $\int \frac{x\, dx}{x^{2}-3x+2}$ , using the identity

\frac{x}{x^{2}-3x+2}=\frac{-1}{x-1}+\frac{2}{x-2}.

Problem 2900

Find the integral $\int \frac{2x+3}{x^{2}-5x+6}\, dx$ , using the identity

\frac{2x+3}{x^{2}-5x+6}=\frac{A}{x-2}+\frac{B}{x-3},

in which the coefficients $A$ and $B$ are easily found by equating coefficients of like powers of $x$ after clearing the denominator.

Problem 2901

Find the integral $\int \frac{dx}{x^{4}-1}$ .

Problem 2902

Find the integral $\int \frac{x\, dx}{x^{3}-1}$ .

Problem 2903

Find the integral $\int \frac{2x^{2}+4}{x^{3}-x^{2}+x-1}\, dx$ .

Problem 2904

Find the integral $\int \frac{dx}{x^{2}(x-1)}$ .

Problem 2905

Find the integral $\int \frac{dx}{(x+a)(x^{2}+b^{2})}$ .

Problem 2906

Find the integral $\int \frac{x^{2}-x+2}{x^{4}-5x^{2}+4}\, dx$ .

Problem 2907

Find the integral $\int \frac{x^{2}\, dx}{(x-1)(x-2)(x-3)}$ .

Problem 2908

Find the integral $\int \frac{x\, dx}{x^{3}-6x^{2}+11x-6}$ .

Problem 2909

Find the integral $\int \frac{dx}{x^{3}+1}$ .

Problem 2910

Find the integral $\int \frac{x^{5}+1}{x^{6}+x^{4}}\, dx$ .

Problem 2911

Find the integral $\int \frac{x^{3}+x^{2}+2}{x(x^{2}-1)^{2}}\, dx$ .

Problem 2912

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

Problem 2913

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

Problem 2914

Find the integral $\int \frac{x\, dx}{x^{3}+1}$ .

Problem 2915

Find the integral $\int \frac{9x^{2}-14x+1}{x^{3}-2x^{2}-x+2}\, dx$ .

Problem 2916

Find the integral $\int \frac{dx}{(x^{2}+9)^{2}}$ .

Problem 2917

Find the integral $\int \frac{dx}{x^{3}+2x-3}$ .

Problem 2918

In the following integrals, to factor the denominator it is convenient to add and subtract the term $ax^{2}$ , where $a$ is a suitably chosen coefficient. Find the integral $\int \frac{dx}{x^{4}+1}$ .

Problem 2919

Find the integral $\int \frac{x^{2}\, dx}{x^{4}+x^{2}+1}$ .

Problem 2920

Find the integral $\int \frac{(x^{2}+2)\, dx}{x^{4}+x^{2}+4}$ .

Problem 2921

Find the integral $\int \frac{dx}{x^{4}+x^{2}+1}$ .

Problem 2922

If the denominator factors into simple factors of the first degree, then for decomposing the fraction into partial fractions the formula

\frac{\varphi (x)}{\psi (x)}=\sum _{k=1}^{n}\frac{\varphi (a_{k})}{\psi '(a_{k})}\frac{1}{x-a_{k}},

is convenient, where $a_{1}, a_{2}, \ldots , a_{n}$ are the roots of the polynomial $\psi (x)$ . If among the roots there is a pair of complex conjugates, then in the sum on the right-hand side the two corresponding terms can be replaced by a single quantity:

\frac{\left(\frac{\varphi (a)}{\psi '(a)}+\frac{\varphi (\bar{a})}{\psi '(\bar{a})}\right)x-\left(\bar{a}\frac{\varphi (a)}{\psi '(a)}+a\frac{\varphi (\bar{a})}{\psi '(\bar{a})}\right)}{x^{2}-(a+\bar{a})x+a\bar{a}}.

Here $\bar{a}=\alpha -\beta i$ is the number complex conjugate to $a=\alpha +\beta i$ .

Find the integral $\int \frac{6\, dx}{x(x-1)(x-2)(x-3)}$ .

Problem 2923

Find the integral $\int \frac{24\, dx}{x(x^{2}-1)(x^{2}-4)}$ .

Problem 2924

Find the integral $\int \frac{dx}{x^{2n}+1}$ .

Problem 2925

The following integrals are reduced to integrals of the previous types after extracting the integer part of the integrand fraction. Find the integral $\int \frac{x^{4}\, dx}{x^{2}+3}$ .

Problem 2926

Find the integral $\int \frac{x^{6}\, dx}{x^{2}-1}$ .

Problem 2927

Find the integral $\int \frac{x^{3}\, dx}{x^{2}+x+1}$ .

Problem 2928

Find the integral $\int \frac{x^{3}\, dx}{x^{2}-3x+2}$ .

Problem 2929

Find the integral $\int \frac{x^{6}+16}{x^{4}-4}\, dx$ .

Problem 2930

Find the integral $\int \frac{x^{4}\, dx}{x^{3}-6x^{2}+11x-6}$ .

Problem 2931

Find the integral $\int \frac{x^{2}+x+1}{x^{2}-x+1}\, dx$ .

Problem 2932

Find the integral $\int \frac{x^{4}+1}{x^{3}-x^{2}+x-1}\, dx$ .

Problem 2933

Find the integral $\int \frac{x^{6}\, dx}{(x^{2}-1)^{2}}$ .

Problem 2934

Using the identity $4a(ax^{2}+bx+c)-(2ax+b)^{2}=4ac-b^{2}$ and integrating by parts the integral $\int \frac{(2ax+b)^{2}\, dx}{(ax^{2}+bx+c)^{n}}$ , one can prove the reduction formula

(4ac-b^{2})u_{n}=\frac{2n-3}{n-1}\cdot 2a u_{n-1}+\frac{1}{n-1}\cdot \frac{2ax+b}{(ax^{2}+bx+c)^{n-1}},

where $u_{n}=\int \frac{dx}{(ax^{2}+bx+c)^{n}}$ .

Find the following integrals using the reduction formula. Find the integral $\int \frac{dx}{(x^{2}+1)^{3}}$ .

Problem 2935

Find the integral $\int \frac{dx}{(x^{2}+x+1)^{3}}$ .

Problem 2936

Find the integral $\int \frac{dx}{(x^{2}+4)^{4}}$ .

Problem 2937

Find the integral $\int \frac{dx}{(x^{2}-3x+3)^{2}}$ .

Problem 2938

Prove the equality

\begin{split} \int \frac{dx}{(x^{2}+a^{2})^{n}} & = -\frac{1}{2n-2}\frac{x}{a^{2}(x^{2}+a^{2})^{n-1}}+\frac{2n-3}{(2n-2)(2n-4)}\frac{x}{a^{4}(x^{2}+a^{2})^{n-2}}+\ldots \\ & \quad \ldots +\frac{(2n-3)(2n-5)\ldots 3\cdot 1}{(2n-2)(2n-4)\ldots 4\cdot 2}\frac{1}{a^{2n-1}}\arctg \frac{x}{a}+C. \end{split}

Problem 2939

Prove the equality

\int \frac{(2ax+b)^{m}}{(ax^{2}+bx+c)^{n}}\, dx=\frac{(2ax+b)^{m-1}}{(n-1)(ax^{2}+bx+c)^{n-1}}+\frac{2a(m-1)}{n-1}\int \frac{(2ax+b)^{m-2}}{(ax^{2}+bx+c)^{n-1}}\, dx.

Problem 2940

Find the integrals. Find the integral $\int \frac{x^{4}\, dx}{(x^{2}+3)^{3}}$ .

Problem 2941

Find the integral $\int \frac{x^{6}\, dx}{(x^{2}-5)^{4}}$ .

Problem 2942

Find the integral $\int \frac{x^{6}\, dx}{(x^{2}+4)^{4}}$ .

Problem 2943

Find the integral $\int \frac{x^{4}\, dx}{(x^{2}-3)^{3}}$ .

Problem 2944

In the following problems the integrand has the form $x^{m}(ax^{n}+b)^{-p}$ . For these the substitution $x^{\sigma }=t$ is useful, where $\sigma$ is the greatest common divisor of $m+1$ and $n$ . Find the integral $\int \frac{x^{5}\, dx}{x^{4}+1}$ .

Problem 2945

Find the integral $\int \frac{x^{3}\, dx}{(x^{8}+4)^{2}}$ .

Problem 2946

Find the integral $\int \frac{x^{3}\, dx}{x^{8}+1}$ .

Problem 2947

Find the integral $\int \frac{x^{5}\, dx}{(x^{12}+1)^{2}}$ .

Problem 2948

Find the integral $\int \frac{dx}{x(x^{3}+4)}$ .

Problem 2949

Find the integral $\int \frac{x^{5}\, dx}{(x^{2}+2)^{2}}$ .

Problem 2950

Find the integral $\int \frac{x^{3}+1}{x(x^{3}-8)}\, dx$ .

Problem 2951

Find the integral $\int \frac{x^{8}-1}{x(x^{8}+1)}\, dx$ .

Problem 2952

Find the integral $\int \frac{dx}{x(x^{6}+1)^{2}}$ .

Problem 2953

Find the integral $\int \frac{x^{5}\, dx}{x^{12}-1}$ .

Problem 2954

In the following problems the integral is found after the denominator is factored. Find the integral $\int \frac{x^{3}+4x^{2}+6x}{x^{4}+2x^{3}+3x^{2}+4x+2}\, dx$ .

Problem 2955

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

Problem 2956

Find the integral $\int \frac{4x^{3}-x^{2}+10x+5}{x^{4}-x^{3}+4x^{2}+3x+5}\, dx$ .

Problem 2957

Find the integral $\int \frac{4x^{3}-17x^{2}+28x-15}{x^{4}-7x^{3}+20x^{2}-27x+15}\, dx$ .

Problem 2958

Find the algebraic part of the integrals. Find the integral $\int \frac{x^{2}+1}{x(x^{3}+1)^{2}}\, dx$ .

Problem 2959

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

Problem 2960

Find the integral $\int \frac{2-3x+x^{2}}{(x+1)^{2}(x^{2}+x+1)^{2}}\, dx$ .

Problem 2961

Find the integral $\int \frac{4x^{5}-1}{(x^{5}+x+1)^{2}}\, dx$ .

Problem 2962

A number of the following problems are solved by various special techniques that avoid decomposing the fraction into partial fractions. The linear-fractional substitution $\frac{x-a}{x-b}=t$ conveniently gives the integral $\int \frac{dx}{(x+1)^{2}(x-2)^{3}}$ .

Problem 2963

Find the integral $\int \frac{dx}{(x^{2}-x)^{3}}$ .

Problem 2964

Find the integral $\int \frac{dx}{(x^{2}-1)^{4}}$ .

Problem 2965

Find the integral $\int \frac{dx}{(x-1)^{2}(x^{2}-1)^{3}}$ .

Problem 2966

The substitutions $x+\frac{1}{x}=u$ and $x-\frac{1}{x}=v$ make it easy to find the following integrals, which contain palindromic polynomials and the factor $x^{2}-1$ or $x^{2}+1$ . Find the integral $\int \frac{x^{2}-1}{x^{4}+3x^{3}+5x^{2}+3x+1}\, dx$ .

Problem 2967

Find the integral $\int \frac{x^{2}-1}{x^{2}+1}\frac{dx}{x^{2}+x+1}$ .

Problem 2968

Find the integral $\int \frac{x^{2}+1}{x^{4}+2x^{3}+3x^{2}-2x+1}\, dx$ .

Problem 2969

Find the integral $\int \frac{x^{2}+1}{x^{4}+x^{2}+1}\, dx$ .

Problem 2970

Find the integral $\int \frac{dx}{x^{4}+x^{2}+1}$ , using the identities

1=\frac{1}{2}(x^{3}+1)-\frac{1}{2}(x^{3}-1), \qquad 1=\frac{1}{2}(x^{2}+1)-\frac{1}{2}(x^{2}-1).

Problem 2971

Find the integral $\int \frac{x^{4}+1}{x^{6}+1}\, dx$ , by subtracting and adding the quantity $x^{2}$ in the numerator.

Problem 2972

Find the integral $A=\int \frac{dx}{x^{4}+1}$ , introducing the auxiliary integral $B=\int \frac{x^{2}\, dx}{x^{4}+1}$ and noting that the integrals $A+B$ and $A-B$ are easily found after the substitutions $x-\frac{1}{x}=u$ and $x+\frac{1}{x}=v$ .

Problem 2973

Find the value of the integral $A=\int \frac{dx}{x^{6}+1}$ , introducing the integral $B=\int \frac{x^{4}\, dx}{x^{6}+1}$ .

Problem 2974

Find the integrals $C=\int \frac{x^{2}\, dx}{x^{8}+1}$ and $D=\int \frac{x^{4}\, dx}{x^{8}+1}$ .

Section 7.3Integration of Irrational Functions[80/100%]

Problem 2975

If the integrand contains a root of some degree of a linear-fractional function $\frac{\alpha x+\beta }{\gamma x+\delta }$ , then this irrationality can be removed by the substitution $\frac{\alpha x+\beta }{\gamma x+\delta }=t^{n}$ , where $n$ is a suitably chosen exponent. Using this technique, find the integral $\int \frac{dx}{\sqrt{2x-1}-\sqrt[3]{2x-1}}$ .

Problem 2976

Find the integral $\int \frac{x\, dx}{(3x-1)\sqrt{3x-1}}$ .

Problem 2977

Find the integral $\int \sqrt{\frac{1-x}{1+x}}\frac{dx}{x}$ .

Problem 2978

Find the integral $\int \frac{x\, dx}{\sqrt{x+1}-\sqrt[3]{x+1}}$ .

Problem 2979

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

Problem 2980

Find the integral $\int \frac{2+\sqrt[3]{x}}{\sqrt[6]{x}+\sqrt[3]{x}+\sqrt{x}+1}\, dx$ .

Problem 2981

The following four integrals, although of a different type, are also found by a substitution of the form $\frac{x-a}{x-b}=t^{n}$ . Find the integral $\int \frac{dx}{\sqrt{(x-1)^{3}(x-2)}}$ .

Problem 2982

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

Problem 2983

Find the integral $\int \frac{dx}{\sqrt[3]{(x+1)^{2}(x-1)^{4}}}$ .

Problem 2984

Find the integral $\int \frac{dx}{\sqrt[3]{(x-1)^{7}(x+1)^{2}}}$ .

Problem 2985

The following three integrals are found by similar techniques. Find the integral $\int \frac{\sqrt[3]{1+\sqrt[4]{x}}}{\sqrt{x}}\, dx$ .

Problem 2986

Find the integral $\int \sqrt[3]{\frac{(x+1)^{5}}{(x-1)^{2}}}\, dx$ .

Problem 2987

Find the integral $\int \frac{dx}{(\sqrt{x+2}+1)\sqrt{\sqrt{x+2}-1}}$ .

Problem 2988

The following integrals have the form $\int x^{m}(ax^{n}+b)^{p}\, dx$ , where $m$ , $n$ , and $p$ are rational numbers. They reduce to integrals of rational functions in three cases: 1) if $p$ is an integer, even if negative (substitution $x=t^{N}$ , where $N$ is the common denominator of the fractions $m$ and $n$ ); 2) if $\frac{m+1}{n}$ is an integer (substitution $ax^{n}+b=t^{M}$ , where $M$ is the denominator of $p$ ); 3) if $\frac{m+1}{n}+p$ is an integer (substitution $a+bx^{-n}=t^{M}$ , where $M$ is the denominator of $p$ ). If none of the three numbers $p$ , $\frac{m+1}{n}$ , and $\frac{m+1}{n}+p$ is an integer, then by Chebyshev's theorem integrals of this form cannot be expressed by a finite combination of elementary functions.

Compute the integral $\int \frac{dx}{\sqrt{x}(\sqrt[3]{x}+1)^{2}}$ .

Problem 2989

Compute the integral $\int \frac{dx}{\sqrt[4]{x^{2}}(\sqrt[3]{x}+1)^{3}}$ .

Problem 2990

Find the integral $\int \frac{\sqrt[3]{x}\, dx}{\sqrt[3]{\sqrt{x}+1}}$ .

Problem 2991

Find the integral $\int \frac{x^{7}\, dx}{\sqrt{x^{2}-1}}$ .

Problem 2992

Find the integral $\int \frac{x^{5}\, dx}{\sqrt[3]{(x^{2}-1)^{2}}}$ .

Problem 2993

Find the integral $\int \frac{dx}{x^{6}\sqrt{x^{2}-1}}$ .

Problem 2994

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

Problem 2995

Find the integral $\int x^{7}\sqrt{1+x^{2}}\, dx$ .

Problem 2996

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

Problem 2997

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

Problem 2998

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

Problem 2999

Find the integral $\int \frac{x^{2}\, dx}{\sqrt{x^{6}-1}}$ .

Problem 3000

The following integrals are solved by a technique similar to that used in the third case of the previous type. Find the integral $\int \frac{dx}{(5x^{3}+3)\sqrt[3]{4x^{3}+3}}$ .

Problem 3001

Find the integral $\int \frac{dx}{(x^{4}+3)\sqrt[3]{2x^{4}+1}}$ .

Problem 3002

In many cases, when finding integrals of differential binomials, i.e., integrals of the form $u_{m,p}=\int x^{m}(ax^{n}+b)^{p}\, dx$ , it is useful first to reduce the integral to another one in which the quantities $m$ and $p$ have more convenient values. This can be done using reduction formulas. The main ones have the following form:

a(m+np+1)u_{m,p}=x^{m-n+1}(ax^{n}+b)^{p+1}-b(m-n+1)u_{m-n,p},

(m+np+1)u_{m,p}=x^{m+1}(ax^{n}+b)^{p}+bnpu_{m,p-1}.

Using the reduction formulas, find the integral $\int \frac{x^{6}\, dx}{\sqrt{x^{2}+1}}$ .

Problem 3003

Find the integral $\int \frac{x^{8}\, dx}{\sqrt{x^{2}-1}}$ .

Problem 3004

Find the integral $\int \frac{dx}{x^{7}\sqrt{x^{4}+1}}$ .

Problem 3005

Find the integral $\int \frac{dx}{(x^{2}+1)^{3}\sqrt{x^{2}+1}}$ .

Problem 3006

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

Problem 3007

Find the integral $\int \frac{x^{9}\, dx}{\sqrt[3]{x^{3}+1}}$ .

Problem 3008

A large number of the following problems give integrals containing the square root of a quadratic polynomial. All of them can be reduced to integrals of rational functions by one of the three Euler substitutions:

\sqrt{ax^{2}+bx+c}=x\sqrt{a}+t, \qquad \sqrt{ax^{2}+bx+c}=\sqrt{c}+tx, \qquad \sqrt{ax^{2}+bx+c}=t(x-x_{1}),

where $x_{1}$ is a root of the polynomial $ax^{2}+bx+c$ .

However, in most cases the Euler substitutions lead to longer computations than direct methods. To apply them one should become familiar with three basic types of the integrals under study.

I. Integrals of the form $\int \frac{f(x)\, dx}{\sqrt{ax^{2}+bx+c}}$ , where $f(x)$ is a polynomial. In most cases the method of undetermined coefficients is convenient. Here we write the presumed identity

\int \frac{f(x)\, dx}{\sqrt{ax^{2}+bx+c}}=(A_{1}x^{n-1}+A_{2}x^{n-2}+\ldots +A_{n})\sqrt{ax^{2}+bx+c}+A_{n+1}\int \frac{dx}{\sqrt{ax^{2}+bx+c}}.

Differentiating both sides of the equality, we obtain

\begin{split} \frac{f(x)}{\sqrt{ax^{2}+bx+c}}={}& \left((n-1)A_{1}x^{n-2}+(n-2)A_{2}x^{n-3}+\ldots +A_{n-1}\right)\sqrt{ax^{2}+bx+c}\ + \\ & +(A_{1}x^{n-1}+A_{2}x^{n-2}+\ldots +A_{n})\frac{2ax+b}{2\sqrt{ax^{2}+bx+c}}+\frac{A_{n+1}}{\sqrt{ax^{2}+bx+c}}. \end{split}

Clearing the denominator, we obtain polynomials of degree $n$ on both sides of the equality. Equating coefficients of like powers of $x$ , we obtain equations from which the coefficients $A_{1}, A_{2}, \ldots , A_{n}$ are found one after another.

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

Problem 3009

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

Problem 3010

Find the integral $\int \frac{x^{3}+2x^{2}+x-1}{\sqrt{x^{2}+2x-1}}\, dx$ .

Problem 3011

Find the integral $\int \frac{x^{3}-x+1}{\sqrt{x^{2}+2x+2}}\, dx$ .

Problem 3012

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

Problem 3013

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

Problem 3014

The preceding problems contained integrals of the form $\int \frac{f(x)\, dx}{\sqrt{ax^{2}+bx+c}}$ , where $f(x)$ is a polynomial. In order to be able to integrate the most general integral of the form $\int f(x,y)\, dx$ , where $f(x,y)$ is a rational function of $x$ and $y$ , and $y=\sqrt{ax^{2}+bx+c}$ , it is important to be able to find integrals of two further types, which we denote by the numerals II and III.

II. Integrals of the form $\int \frac{f_{m}(x)\, dx}{(x-\alpha )^{n}\sqrt{ax^{2}+bx+c}}$ , where $f_{m}(x)$ is a polynomial of degree $m$ less than $n$ . These integrals reduce to integrals of the previous class by the substitution $x-\alpha =\frac{1}{t}$ . If the degree of the polynomial $f_{m}(x)$ is not less than $n$ , then the integer part of the fraction $\frac{f_{m}(x)}{(x-\alpha )^{n}}$ should be extracted first.

Using these methods, find the integral $\int \frac{dx}{x\sqrt{x^{2}-1}}$ .

Problem 3015

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

Problem 3016

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

Problem 3017

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

Problem 3018

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

Problem 3019

Find the integral $\int \frac{x^{3}\, dx}{(x-1)^{2}\sqrt{x^{2}+2x+4}}$ .

Problem 3020

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

Problem 3021

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

Problem 3022

III. The third and most complicated type is represented by integrals

I=\int \frac{(Mx+N)\, dx}{(x^{2}+px+q)^{n}\sqrt{ax^{2}+bx+c}},

where the roots of the trinomial $x^{2}+px+q$ are imaginary. If $p=b=0$ , then

I=M\int \frac{x\, dx}{(x^{2}+q)^{n}\sqrt{ax^{2}+c}}+N\int \frac{dx}{(x^{2}+q)^{n}\sqrt{ax^{2}+c}}=M\int \frac{x\, dx}{(x^{2}+q)^{n}\sqrt{ax^{2}+c}}+N\int \frac{x^{-2n-1}\, dx}{(1+qx^{-2})^{n}\sqrt{a+cx^{-2}}}.

Here the first integral is found by the substitution $ax^{2}+c=u^{2}$ , and the second by the substitution $a+cx^{-2}=v^{2}$ . In the general case the integral $I$ is reduced to this simplest case by the substitution $x=\frac{\alpha t+\beta }{t+1}$ . This gives

I=\int \frac{f_{n}(t)\, dt}{(At^{2}+Bt+C)^{n}\sqrt{A_{1}t^{2}+B_{1}t+C_{1}}},

where $f_{n}(t)$ is a polynomial of degree $n$ , and the coefficients $B$ and $B_{1}$ have the following values:

B=2\alpha \beta +p(\alpha +\beta )+2q; \qquad B_{1}=2a\alpha \beta +b(\alpha +\beta )+2c.

Setting $B=0$ , $B_{1}=0$ , we obtain equations giving real values for $\alpha$ and $\beta$ for real $p, q, a, b, c$ , provided at least one of the polynomials has imaginary roots. With this choice of the numbers $\alpha$ and $\beta$ , the integral $I$ splits into a sum of integrals of two types:

\int \frac{t^{2\nu +1}\, dt}{(At^{2}+C)\sqrt{A_{1}t^{2}+C_{1}}} \quad \text{and} \quad \int \frac{t^{2\nu }\, dt}{(At^{2}+C)\sqrt{A_{1}t^{2}+C_{1}}},

which are integrated respectively by the substitutions:

A_{1}t^{2}+C_{1}=\sigma ^{2} \quad \text{and} \quad A_{1}+C_{1}t^{-2}=\tau ^{2}.

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

Problem 3023

Find the integral $\int \frac{(x+3)\, dx}{(x^{2}+1)\sqrt{x^{2}+x+1}}$ .

Problem 3024

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

Problem 3025

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

Problem 3026

Find the integral $\int \frac{(2x+1)\, dx}{(3x^{2}+4x+4)\sqrt{x^{2}+6x-1}}$ .

Problem 3027

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

Problem 3028

The most general integral of the form $\int f(x,\sqrt{ax^{2}+bx+c})\, dx$ , where $f(x,y)$ is a rational function of $x$ and $y$ , reduces to integrals of the three previous types. For this it suffices, by rationalizing the denominator, to transform the function $f(x,y)$ to the form $\varphi (x)+\psi (x)y$ , where $\varphi (x)$ and $\psi (x)$ are rational functions of $x$ . Here $\psi (x)y=\frac{\psi (x)y^{2}}{y}=\frac{\psi (x)(ax^{2}+bx+c)}{y}=\frac{\omega (x)}{\sqrt{ax^{2}+bx+c}}$ . Thus the matter reduces to finding the integral of the rational function $\varphi (x)$ and of $\frac{\omega (x)}{\sqrt{ax^{2}+bx+c}}$ , where $\omega (x)$ is a rational fraction. In the latter integral the rational fraction $\omega (x)$ can be transformed by extracting the integer part and decomposing into partial fractions. This yields integrals of the three types already considered. In simple cases part of these transformations is unnecessary.

Find the integral $\int \sqrt{\frac{a-x}{x-b}}\, dx$ .

Problem 3029

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

Problem 3030

Find the integral $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\, dx$ .

Problem 3031

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

Problem 3032

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

Problem 3033

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

Problem 3034

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

Problem 3035

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

Problem 3036

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

Problem 3037

Find the integral $\int \frac{x^{3}\, dx}{(x^{2}-1)\sqrt{x^{2}+2x+4}}$ .

Problem 3038

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

Problem 3039

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

Problem 3040

Find the integral $\int \frac{dx}{(x^{2}+x-2)\sqrt{x^{2}+2x+3}}$ .

Problem 3041

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

Problem 3042

The substitution $\frac{2ax+b}{\sqrt{ax^{2}+bx+c}}=u$ is sometimes called the Abel substitution; it is convenient for finding the integral of the function $(ax^{2}+bx+c)^{-n-\frac{1}{2}}$ , where $n$ is an integer.

Find the integral $\int \frac{dx}{\sqrt{(2x-x^{2})^{3}}}$ .

Problem 3043

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

Problem 3044

Find the integral $\int \frac{(3x+2)\, dx}{(x^{2}+4x+1)^{2}\sqrt{x^{2}+4x+1}}$ .

Problem 3045

Find the integral $\int \frac{(x^{3}+1)\, dx}{(x^{2}+x+1)\sqrt{x^{2}+x+1}}$ .

Problem 3046

Integrals $\int f(x,y)\, dx$ , where $f(x,y)$ is a rational function and $y=\sqrt{Ax^{4}+Bx^{3}+Cx^{2}+Dx+E}$ , called elliptic integrals, generally cannot be integrated in closed form unless there are special relations among the coefficients of the function $f(x,y)$ or of the polynomial $Ax^{4}+Bx^{3}+Cx^{2}+Dx+E$ .

In the following examples with palindromic polynomials, one of the substitutions $x+\frac{1}{x}=u$ or $x-\frac{1}{x}=v$ makes it comparatively easy to obtain the integral. Find the integral $\int \frac{x^{2}-1}{x^{2}+1}\frac{dx}{\sqrt{x^{4}+1}}$ .

Problem 3047

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

Problem 3048

Find the integral $\int \frac{x^{2}-1}{x\sqrt{1+3x^{2}+x^{4}}}\, dx$ .

Problem 3049

Find the integral $\int \frac{(x^{2}+1)\, dx}{x\sqrt{x^{4}+3x^{3}-2x^{2}-3x+1}}$ .

Problem 3050

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

Problem 3051

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

Problem 3052

Theorem (Abel). Let $y$ be an algebraic function of $x$ of degree $n$ , i.e., $y$ satisfies the equation $\varphi (x,y)=0$ , where $\varphi (x,y)$ is a polynomial in $x$ and $y$ of degree $n$ in $y$ . If the integral $\int f(x,y)\, dx$ of the function $f(x,y)$ , rational in $x$ and $y$ , can be expressed through algebraic functions, then the equality

\int f(x,y)\, dx=p_{0}(x)+p_{1}(x)y+\ldots +p_{n-1}(x)y^{n-1},

must hold, where $p_{0}(x), p_{1}(x), \ldots , p_{n-1}(x)$ are rational functions of $x$ .

Corollary. If $y=\sqrt[m]{\varphi (x)}$ , where $\varphi (x)$ is a polynomial of degree $n$ , and if the integral $\int \frac{a_{0}x^{\lambda }+a_{1}x^{\lambda -1}+\ldots +a_{\lambda }}{y^{k}}\, dx$ is expressed by an algebraic function, then it must equal $f(x)y^{m-k}$ , where $f(x)$ is a polynomial of degree $\lambda =n+1$ .

Find the following algebraic integrals. Find the integral $\int \frac{3x^{4}+2x^{3}+8x^{2}+6x+2}{\sqrt[3]{(x^{3}+3x+1)^{2}}}\, dx$ .

Problem 3053

Find the integral $\int \frac{5x^{5}+11x^{3}+3x^{2}}{\sqrt[3]{x^{3}+3x+1}}\, dx$ .

Problem 3054

Find the integral $\int \frac{5x^{7}+17x^{4}+4x^{3}}{\sqrt[4]{(x^{4}+4x+1)^{3}}}\, dx$ .

Section 7.4Integration of Transcendental Functions[108/97%]

Problem 3055

Integrals containing the functions $\ln \varphi$ , $\arctg \varphi$ , $\arcsin \varphi$ , where $\varphi$ is some algebraic function of $x$ , are often integrated by parts, setting $u=\ln \varphi$ , $u=\arctg \varphi$ , $u=\arcsin \varphi$ . This is how the integral $\int \ln x\, dx$ is found.

Problem 3056

Find the integral $\int \frac{\ln x}{(2x+5)^{3}}\, dx$ .

Problem 3057

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

Problem 3058

Find the integral $\int \frac{\ln (x-1)}{(x+1)^{3}}\, dx$ .

Problem 3059

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

Problem 3060

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

Problem 3061

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

Problem 3062

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

Problem 3063

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

Problem 3064

Find the integral $\int \arctg x\, dx$ .

Problem 3065

Find the integral $\int x^{6}\arctg x\, dx$ .

Problem 3066

Find the integral $\int x^{7}\arctg x\, dx$ .

Problem 3067

Find the integral $\int \arcsin x\, dx$ .

Problem 3068

Find the integral $\int x\arcsin x\, dx$ .

Problem 3069

Find the integral $\int \frac{\arcsin x}{x^{2}}\, dx$ .

Problem 3070

Find the integral $\int \frac{x\arcsin x\, dx}{(1-x^{2})\sqrt{1-x^{2}}}$ .

Problem 3071

Integrals containing the product of a polynomial with one of the functions $e^{ax}$ , $\cos ax$ , $\sin bx$ are best evaluated using the formula for repeated integration by parts, which can be given the form

\int uv^{(n)}\, dx=uv^{(n-1)}-u'v^{(n-2)}+u''v^{(n-3)}-\ldots +(-1)^{m}u^{(m)}v^{(n-m-1)}+(-1)^{(m+1)}\int u^{(m+1)}v^{(n-m-1)}\, dx+C.

If $u$ is a polynomial of degree $m$ , then the last term vanishes.

Prove the equalities, in which $f(x)$ is a polynomial of degree $m$ :

\int f(x)e^{ax}\, dx=\frac{e^{ax}}{a}\left(f(x)-\frac{f'(x)}{a}+\frac{f''(x)}{a^{2}}-\ldots +(-1)^{m}\frac{f^{(m)}(x)}{a^{m}}\right)+C.

Problem 3072

Prove the equality

\begin{split} \int f(x)\cos ax\, dx={}& \frac{\sin ax}{a}\left(f(x)-\frac{f''(x)}{a^{2}}+\frac{f^{IV}(x)}{a^{4}}-\ldots \right)\ + \\ & +\frac{\cos ax}{a^{2}}\left(f'(x)-\frac{f'''(x)}{a^{2}}+\frac{f^{V}(x)}{a^{4}}-\ldots \right)+C. \end{split}

Problem 3073

Prove the equality

\begin{split} \int f(x)\sin ax\, dx={}& -\frac{\cos ax}{a}\left(f(x)-\frac{f''(x)}{a^{2}}+\frac{f^{IV}(x)}{a^{4}}-\ldots \right)\ + \\ & +\frac{\sin ax}{a^{2}}\left(f'(x)-\frac{f'''(x)}{a^{2}}+\frac{f^{V}(x)}{a^{4}}-\ldots \right)+C. \end{split}

Problem 3074

Find the integral $\int x^{4}e^{-x}\, dx$ .

Problem 3075

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

Problem 3076

Find the integral $\int x^{3}\sin (2x+3)\, dx$ .

Problem 3077

Find the integral $\int x^{4}\sin x\, dx$ .

Problem 3078

Find the integral $\int (x^{3}-2x^{2}+5)e^{3x}\, dx$ .

Problem 3079

Find the integral $\int (x^{2}+3x+5)\cos 2x\, dx$ .

Problem 3080

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

Problem 3081

Find the integral $\int (x^{3}-x^{2}+x)\sin x\, dx$ .

Problem 3082

The following integrals, containing exponential functions, are found by integration by parts, and also by simple substitutions. Find the integral $\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\, dx$ .

Problem 3083

Find the integral $\int \frac{dx}{\sqrt{e^{x}+1}}$ .

Problem 3084

Find the integral $\int \frac{dx}{\sqrt{e^{2x}+e^{x}+1}}$ .

Problem 3085

Find the integral $\int x^{5}e^{-x^{2}}\, dx$ .

Problem 3086

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

Problem 3087

Find the integral $\int \frac{xe^{x}}{(x+1)^{2}}\, dx$ .

Problem 3088

Find the integral $\int \frac{x^{3}+2x^{2}+3x+3}{x^{4}}e^{-x}\, dx$ .

Problem 3089

Find the integral $\int \frac{x^{3}+2x^{2}-2x-2}{(x^{2}+x)^{2}}e^{x}\, dx$ .

Problem 3090

The following integrals are most conveniently found by using the fact that the integration formulas for functions containing $e^{ax}$ remain valid for complex values of $a$ . Separating the real and imaginary parts, we obtain equalities with real integrals. This is how the integral $\int e^{ax}\cos bx\, dx$ is found.

Problem 3091

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

Problem 3092

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

Problem 3093

Find the integral $\int x^{2}\cos x\, e^{x}\, dx$ .

Problem 3094

A number of the following integrals belong to the type $\int f(\cos x, \sin x)\, dx$ , where $f(u,v)$ is a rational function. Here it is useful to keep in mind three rules: 1) if $f(\cos x, \sin x)$ is multiplied by $-1$ when the sign is changed before one of the quantities $\cos x$ or $\sin x$ , then it is useful to denote the other of them by $t$ ; 2) if $f(\cos x, \sin x)$ does not change when the sign is changed simultaneously before $\cos x$ and $\sin x$ , then it is useful to set $\tg x=t$ or $\ctg x=t$ ; 3) the substitution $\tg \frac{x}{2}=t$ in all cases turns an integral of the type under consideration into an integral of a rational fraction. Unfortunately, this substitution leads to longer computations than the previous ones, when the latter are applicable.

Find the integral $\int \sin^{3}x\, dx$ .

Problem 3095

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

Problem 3096

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

Problem 3097

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

Problem 3098

Find the integral $\int \frac{dx}{\sin^{3}x}$ .

Problem 3099

Find the integral $\int \frac{dx}{\cos^{5}x}$ .

Problem 3100

Find the integral $\int \tg^{6}x\, dx$ .

Problem 3101

Find the integral $\int \ctg^{3}x\, dx$ .

Problem 3102

Find the integral $\int \frac{dx}{\cos^{4}x}$ .

Problem 3103

Find the integral $\int \sin^{7}x\cos^{6}x\, dx$ .

Problem 3104

Find the integral $\int \frac{\cos^{5}x}{\sin^{3}x}\, dx$ .

Problem 3105

Find the integral $\int \frac{\cos^{3}x}{\sin^{5}x}\, dx$ .

Problem 3106

Find the integral $\int \frac{dx}{\sin x\cos^{3}x}$ .

Problem 3107

Find the integral $\int \frac{dx}{\sin^{4}x\cos^{4}x}$ .

Problem 3108

For the integrals $U_{m,n}=\int \sin^{m}x\cos^{n}x\, dx$ there exist reduction formulas that allow reducing the integrals to others in which the indices $m$ and $n$ are closer to zero. These formulas reduce to four basic ones:

(m+n)U_{m,n}=-\sin ^{m-1}x\cos ^{n+1}x+(m-1)U_{m-2,n};

(m+n)U_{m,n}=\sin ^{m+1}x\cos ^{n-1}x+(n-1)U_{m,n-2};

(n+1)U_{m,n}=-\sin ^{m-1}x\cos ^{n+1}x+(m-1)U_{m-2,n+2};

(m+1)U_{m,n}=\sin ^{m+1}x\cos ^{n-1}x+(n-1)U_{m+2,n-2}.

Denoting by $U_{n}$ the integral $U_{0,n}$ and by $V_{m}$ the integral $U_{m,0}$ , we obtain from this two more reduction formulas for the integrals $U_{n}=\int \cos^{n}x\, dx$ and $V_{m}=\int \sin^{m}x\, dx$ :

mV_{m}=-\sin ^{m-1}x\cos x+(m-1)V_{m-2}, \qquad nU_{n}=\sin x\cos ^{n-1}x+(n-1)U_{n-2}.

Using the reduction formulas, find the integral $\int \sin^{4}x\cos^{2}x\, dx$ .

Problem 3109

Find the integral $\int \sin^{6}x\cos^{4}x\, dx$ .

Problem 3110

Find the integral $\int \sin^{2}x\cos^{6}x\, dx$ .

Problem 3111

Find the integral $\int \sin^{4}x\cos^{6}x\, dx$ .

Problem 3112

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

Problem 3113

Find the integral $\int \sin^{8}x\, dx$ .

Problem 3114

Find the integral $\int \frac{\cos^{8}x}{\sin^{2}x}\, dx$ .

Problem 3115

Find the integral $\int \frac{\sin^{10}x}{\cos^{4}x}\, dx$ .

Problem 3116

Below is a series of integrals of trigonometric functions found by various techniques. Find the integral $\int \frac{\sin x-\cos x}{\sin x+\cos x}\, dx$ , by comparing the numerator with the derivative of the denominator.

Problem 3117

Find the integral $\int \frac{3\cos x+7\sin x}{5\cos x+2\sin x}\, dx$ , by writing the numerator in the form $Au+Bu'$ , where $u=5\cos x+2\sin x$ .

Problem 3118

Find the integral $\int \frac{dx}{\sin x+\cos x}$ , by reducing the denominator to logarithmic form.

Problem 3119

Apply the same technique to the integral $\int \frac{dx}{a\sin x+b\cos x}$ .

Problem 3120

Find the integral $\int \sqrt{1+\sin x}\, dx$ , using the fact that $1=\cos^{2}\frac{x}{2}+\sin^{2}\frac{x}{2}$ .

Problem 3121

Using the substitution $\tg x=t$ , find the integral $\int \frac{dx}{\sqrt[4]{\sin^{3}x\cos^{5}x}}$ .

Problem 3122

Find the integral $\int \sqrt{\frac{\sin^{3}x}{\cos^{7}x}}\, dx$ .

Problem 3123

Find the integral $\int \frac{dx}{\sqrt[3]{\tg x}}$ .

Problem 3124

Find the integral $\int \sqrt{\tg x}\, dx$ .

Problem 3125

Find the integral $\int \frac{dx}{1+\tg x}$ .

Problem 3126

Find the integral $\int \frac{dx}{4+3\tg x}$ .

Problem 3127

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

Problem 3128

Find the integral $\int \frac{1+\tg^{2}x}{(4+\tg^{2}x)\tg^{3}x}\, dx$ .

Problem 3129

Find the integral $\int \frac{1+\tg x}{\sin 2x}\, dx$ .

Problem 3130

Find the integral $\int \frac{dx}{2+3\cos^{2}x}$ .

Problem 3131

Find the integral $\int \frac{\sin^{2}x\cos^{2}x}{(\cos^{3}x+\sin^{3}x)^{2}}\, dx$ .

Problem 3132

Find the integral $\int \frac{dx}{a^{2}-b^{2}\sin^{2}x}$ ; $a^{2}\neq b^{2}$ .

Problem 3133

Find the integral $\int \frac{2\sqrt{\tg x-1}}{(1+\cos 2x)\sqrt{1+\tg x}}\, dx$ .

Problem 3134

Find the integral $\int \frac{\cos x\, dx}{\sin^{3}x-\cos^{3}x}$ .

Problem 3135

Find the integrals. Find the integral $\int \frac{\sin 2x\, dx}{\sin^{4}x+\cos^{4}x}$ .

Problem 3136

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

Problem 3137

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

Problem 3138

Find the integral $\int \frac{\sin 3x}{\cos x}\, dx$ .

Problem 3139

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

Problem 3140

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

Problem 3141

Find the integral $\int \frac{\cos^{4}x}{\sin 6x}\, dx$ .

Problem 3142

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

Problem 3143

Find the integral $\int \frac{\cos 2x}{\sin^{6}x}\, dx$ .

Problem 3144

Find the integral $\int \frac{\cos 3x}{\cos^{4}x}\, dx$ .

Problem 3145

Find the integral $\int \frac{\cos 3x}{\sin^{5}x}\, dx$ .

Problem 3146

Find the integral $\int \frac{\sin x}{\sqrt{\cos 2x}}\, dx$ .

Problem 3147

Find the integral $\int \frac{\cos x}{\sqrt{\sin 2x}}\, dx$ .

Problem 3148

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

Remark. Here the substitution $\frac{\pi }{4}-x=t$ is useful.

Problem 3149

Find the integral $\int \frac{dx}{a+b\cos x}$ for $a^{2}>b^{2}$ and for $a^{2}<b^{2}$ .

Problem 3150

The following integrals contain hyperbolic functions. In these examples it is convenient to express them through exponential functions. Find the integral $\int \frac{x+\sh x}{\ch x-\sh x}\, dx$ .

Problem 3151

Find the integral $\int \frac{\ch^{3}x-\sh^{3}x}{\ch^{3}x+\sh^{3}x}\, dx$ .

Problem 3152

Find the integral $\int \sqrt{\th x}\, dx$ .

Problem 3153

Find the integral $\int \frac{\th x}{\sqrt{1-\th x}}\, dx$ .

Problem 3154

Find the integral $\int (\ch x+\sh x)\sqrt{\ch x-\sh x}\, dx$ .

Problem 3155

In the following examples it is more convenient not to express the hyperbolic functions through exponentials. Find the integral $\int \frac{dx}{(1-\ch x)^{2}}$ .

Problem 3156

Find the integral $\int \frac{dx}{(1+\ch^{2}x)^{2}}$ .

Problem 3157

Find the integral $\int \frac{\ch^{3}x\, dx}{1-\th x}$ .

Problem 3158

Find the integral $\int \frac{dx}{1-\sh^{4}x}$ .

Problem 3159

Find the integral $\int \frac{dx}{\sh^{3}x}$ .

Problem 3160

Find the integral $\int \frac{\sh^{3}x\, dx}{\sqrt{\ch x}}$ .

Problem 3161

Find the integral $\int \th^{3}x\, dx$ .

Problem 3162

Find the integral $\int \ch x\ch 2x\ch 3x\, dx$ .

Section 7.5Miscellaneous Problems[94/100%]

Problem 3163

Find the integral $\int \frac{x^{2}+x-1}{x^{3}-x^{2}-x+1}\, dx$ .

Problem 3164

Find the integral $\int \frac{x\, dx}{(x-1)^{3}(x-2)^{2}}$ .

Problem 3165

Find the integral $\int \frac{x^{5}\, dx}{(x-1)^{2}(x^{2}-1)}$ .

Problem 3166

Find the integral $\int \frac{x^{2}\, dx}{(x-a)(x-b)(x-c)}$ .

Problem 3167

Find the integral $\int \frac{dx}{x^{5}+2x^{3}+3x^{2}}$ .

Problem 3168

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

Problem 3169

Find the integral $\int \frac{x^{2}(1-x^{2})\, dx}{(1+x^{2})^{3}}$ .

Problem 3170

Find the integral $\int \frac{x^{2}\, dx}{x^{4}+x^{2}-2}$ .

Problem 3171

Find the integral $\int \frac{x\, dx}{(x^{2}+2x+2)^{2}}$ .

Problem 3172

Find the integral $\int \frac{x^{9}\, dx}{(x^{2}-1)^{3}}$ .

Problem 3173

Find the integral $\int \frac{x^{3}+3x}{(x^{2}-1)(x^{2}+1)^{2}}\, dx$ .

Problem 3174

Find the integral $\int x\sqrt{\frac{a-x}{a+x}}\, dx$ .

Problem 3175

Find the integral $\int \frac{x\, dx}{1+\sqrt{x-1}}$ .

Problem 3176

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

Problem 3177

Find the integral $\int \frac{x+\sqrt{x+1}}{1+\sqrt{x+1}}\, dx$ .

Problem 3178

Find the integral $\int \frac{x^{3}\, dx}{\sqrt{2ax-x^{2}}}$ .

Problem 3179

Find the integral $\int \frac{x\, dx}{(a+bx)^{\frac{3}{2}}}$ .

Problem 3180

Find the integral $\int \frac{dx}{x^{2}(a+bx^{2})^{\frac{3}{2}}}$ .

Problem 3181

Find the integral $\int \frac{dx}{x^{3}\sqrt{x-1}}$ .

Problem 3182

Find the integral $\int \frac{dx}{x^{\frac{3}{4}}\left(1+x^{\frac{1}{3}}\right)}$ .

Problem 3183

Find the integral $\int \frac{dx}{x^{2}\sqrt{ax+bx^{4}}}$ .

Problem 3184

Find the integral $\int \frac{x^{3}\, dx}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}$ .

Problem 3185

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

Problem 3186

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

Problem 3187

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

Problem 3188

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

Problem 3189

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

Problem 3190

Find the integral $\int \frac{\sqrt{a+bx^{2}}}{x^{6}}\, dx$ .

Problem 3191

Find the integral $\int \frac{dx}{x^{2}\sqrt{x^{2}-x}}$ .

Problem 3192

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

Problem 3193

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

Problem 3194

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

Problem 3195

Find the integral $\int \frac{x+1}{x-2}\frac{dx}{\sqrt{1+x^{3}}}$ .

Problem 3196

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

Problem 3197

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

Problem 3198

Find the integral $\int \frac{(x^{2}-a^{2})\, dx}{x\sqrt{a^{4}+x^{4}}}$ .

Problem 3199

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

Problem 3200

Find the integral $\int \frac{dx}{\sqrt{(x-2)(2-3x+x^{3})}}$ .

Problem 3201

Find the integral $\int \frac{3x^{2}+5x^{6}}{\sqrt{1+x^{4}}}\, dx$ .

Problem 3202

Find the integral $\int \frac{dx}{\sqrt{x^{2}-a^{2k-2}x^{2k}}}$ .

Problem 3203

Find the integral $\int \frac{(x^{4}-a^{4})\, dx}{(x^{4}+a^{4})\sqrt{x^{4}+a^{4}}}$ .

Problem 3204

Find the integral $\int \frac{1+x^{2}}{1-2x^{2}\cos \alpha +x^{4}}\, dx$ .

Problem 3205

Find the integral $\int \frac{x^{2}\, dx}{(x\sin x+\cos x)^{2}}$ .

Problem 3206

Find the integral $\int \frac{x^{2}\, dx}{(\sin x-x\cos x)^{2}}$ .

Problem 3207

Find the integral $\int \frac{dx}{(a+(ax+b)\tg x)^{2}}$ .

Problem 3208

Find the integral $\int \frac{a+x}{x^{2}+ax\ln x}\, dx$ .

Problem 3209

Find the integral $\int \frac{1-\ln x}{(x+a\ln x)^{2}}\, dx$ .

Problem 3210

Find the integral $\int \frac{1+x\ln x}{x}e^{x}\, dx$ .

Problem 3211

Find the integral $\int \left(\frac{1-x}{1+x^{2}}\right)^{2}e^{x}\, dx$ .

Problem 3212

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

Problem 3213

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

Problem 3214

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

Problem 3215

Find the integral $\int \frac{xe^{x}}{(e^{x}-1)^{3}}\, dx$ .

Problem 3216

Find the integral $\int \frac{dx}{\cos x-2\tg x}$ .

Problem 3217

Find the integral $\int \frac{dx}{\sin^{4}x\cos x}$ .

Problem 3218

Find the integral $\int \frac{dx}{1-\cos^{4}x}$ .

Problem 3219

Find the integral $\int \frac{dx}{\sin x\cos^{2}x}$ .

Problem 3220

Find the integral $\int \frac{\ln \tg x\, dx}{\cos^{4}x}$ .

Problem 3221

Find the integral $\int \frac{\cos^{2}x\, dx}{\sin x(1+a^{2}\cos^{2}x)}$ .

Problem 3222

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

Problem 3223

Find the integral $\int \frac{dx}{\sin x\sqrt{\cos 2x}}$ .

Problem 3224

Find the integral $\int \tg x\sqrt{\cos 2x}\, dx$ .

Problem 3225

Find the integral $\int \frac{dx}{\sqrt{2+3\sin x+\sin^{2}x}}$ .

Problem 3226

Find the integral $\int \frac{\sqrt{\sin x}}{1+\sin^{2}x}\, dx$ .

Problem 3227

Find the integral $\int (b^{2}-a^{2}\tg^{2}x)^{\frac{3}{2}}\, dx$ .

Problem 3228

Find the integral $\int \frac{dx}{\cos x\sqrt{\cos 2x}}$ .

Problem 3229

Find the integral $\int \arctg \sqrt{x}\, dx$ .

Problem 3230

Find the integral $\int \frac{\arctg x}{(1-x^{2})\sqrt{1-x^{2}}}\, dx$ .

Problem 3231

Find the integral $\int e^{a\cdot \arctg x}\frac{x\, dx}{(1+x^{2})\sqrt{1+x^{2}}}$ .

Problem 3232

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

Problem 3233

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

Problem 3234

Find the integral $\int \frac{x+\sin x}{1+\cos x}\, dx$ .

Problem 3235

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

Problem 3236

Find the integral $\int \ln (\sqrt{x-a}+\sqrt{x-b})\, dx$ .

Problem 3237

Find the integral $\int \frac{dx}{2^{x}+1}$ .

Problem 3238

Find the integral $\int \frac{x+1}{2^{x}}\, dx$ .

Problem 3239

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

Problem 3240

Find the integral $\int \frac{\cos x\, dx}{\sqrt{\cos^{3}2x}}$ .

Problem 3241

Find the integral $\int \frac{dx}{\sin x\sqrt[4]{\cos 2x}}$ .

Problem 3242

Find the integral $\int \sin^{n-1}x\cos (n+1)x\, dx$ .

Problem 3243

Find the integral $\int \sin^{n-1}x\sin (n+1)x\, dx$ .

Problem 3244

Find the integral $\int \frac{dx}{a+b\cos x+c\sin x}$ .

Problem 3245

Find the integral $\int \frac{\sqrt{\ctg x}-\sqrt{\tg x}}{1+3\sin 2x}\, dx$ .

Problem 3246

Find the integral $\int x^{\sqrt{2}}\sin \ln x\, dx$ .

Problem 3247

Find the integral $\int \frac{\ln (\cos \varphi +\sqrt{\cos 2\varphi })}{1-\cos^{2}\varphi }\, d\varphi$ .

Problem 3248

Find the integral $\int \frac{1+\sin x}{1+\cos x}e^{x}\, dx$ .

Problem 3249

Find the integral $\int \frac{\sin x-\cos x}{\sin^{2}x}e^{x}\, dx$ .

Problem 3250

Find the integral $\int \frac{\sin x}{\sin (x-\alpha )}\, dx$ .

Problem 3251

Find the integral $\int \frac{a\sin x+b\cos x}{\sqrt{1+\sin x\cos x}}\, dx$ .

Problem 3252

Find the integral $\int \frac{dx}{\cos x\cos (x-\alpha )}$ .

Problem 3253

Find the integral $\int \frac{dx}{\sqrt{a+b+(a+2b)\cos x+b\cos 2x}}$ .

Problem 3254

Find the integral $\int \frac{dx}{\cos^{3}x+\sin^{3}x}$ .

Problem 3255

Find the integral $\int \frac{dx}{\cos^{5}x-\sin^{5}x}$ .

Problem 3256

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