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integrals value
i.e. $$\int e^x dx = e^x$$
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Equation 0:
$$\int x^n dx$$
$$\frac{1}{n-1}x^{n-1},\hspace{1ex}n\neq +1$$
$$\frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1$$

Next Problem

Equation 1:
$$\int \frac{1}{x}dx$$
$$\ln |x|$$
$$\log |x|$$

Next Problem

Equation 2:
$$\int u dv$$
$$uv - \int v du$$
$$uv + \int v du$$

Next Problem

Equation 3:
$$\int \frac{1}{ax+b}dx$$
$$\frac{1}{a} \ln |ax + b|$$
$$\frac{1}{a} \log |ax + b|$$

Next Problem

Equation 4:
$$\int \frac{1}{(x+a)^2}dx$$
$$-\frac{1}{x+a}$$
$$\frac{1}{x-a}$$

Next Problem

Equation 5:
$$\int (x+a)^n dx$$
$$\frac{(x+a)^{n+1}}{n+1}, n\ne -1$$
$$\frac{(x-a)^{n-1}}{n-1}, n\ne +1$$

Next Problem

Equation 6:
$$\int x(x+a)^n dx$$
$$\frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)}$$
$$\frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+3)}$$

Next Problem

Equation 7:
$$\int \frac{1}{1+x^2}dx$$
$$\cos^{-1}x$$
$$\tan^{-1}x$$

Next Problem

Equation 8:
$$\int \frac{1}{a^2+x^2}dx$$
$$\frac{1}{a}\tan^{+1}\frac{x}{a}$$
$$\frac{1}{a}\tan^{-1}\frac{x}{a}$$

Next Problem

Equation 9:
$$\int \frac{x}{a^2+x^2}dx$$
$$\frac{1}{2}\ln|a^2+x^2|$$
$$\frac{1}{2}\ln|a^2-x^2|$$

Next Problem

Equation 10:
$$\int \frac{x^2}{a^2+x^2}dx$$
$$x+a\tan^{+1}\frac{x}{a}$$
$$x-a\tan^{-1}\frac{x}{a}$$

Next Problem

Equation 11:
$$\int \frac{x^3}{a^2+x^2}dx$$
$$\frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2|$$
$$\frac{1}{2}x^2-\frac{1}{2}a^2\log|a^2+x^2|$$

Next Problem

Equation 12:
$$\int \frac{1}{ax^2+bx+c}dx$$
$$\frac{2}{\sqrt{4ac-b^2}}\cos^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}$$
$$\frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}$$

Next Problem

Equation 13:
$$\int \frac{1}{(x+a)(x+b)}dx$$
$$\frac{1}{b-a}\ln\frac{a+x}{b+x}, \text{ } a\ne b$$
$$\frac{1}{b+a}\ln\frac{a-x}{b-x}, \text{ } a\ne b$$

Next Problem

Equation 14:
$$\int \frac{x}{(x+a)^2}dx$$
$$\frac{a}{a+x}+\log |a+x|$$
$$\frac{a}{a+x}+\ln |a+x|$$

Next Problem

Equation 15:
$$\int \frac{x}{ax^2+bx+c}dx$$
$$\frac{1}{2a}\ln|ax^2+bx+c| -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}$$
$$\frac{1}{3a}\ln|ax^3+bx+c| -\frac{b}{a\sqrt{4ac-b^3}}\tan^{-1}\frac{3ax+b}{\sqrt{4ac-b^3}}$$

Next Problem

Equation 16:
$$\int \sqrt{x-a}\ dx$$
$$\frac{2}{3}(x-a)^{3/2}$$
$$\frac{3}{2}(x-a)^{2/3}$$

Next Problem

Equation 17:
$$\int \frac{1}{\sqrt{x\pm a}}\ dx$$
$$2\sqrt{x\pm a}$$
$$3\sqrt{x\pm a}$$

Next Problem

Equation 18:
$$\int \sqrt{ax+b}\ dx$$
$$\left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b}$$
$$\left(\frac{3b}{2a}+\frac{3x}{2}\right)\sqrt{ax+b}$$

Next Problem

Equation 19:
$$\int (ax+b)^{3/2}\ dx$$
$$\frac{2}{5a}(ax+b)^{5/2}$$
$$\frac{2}{5a}(ax-b)^{5/2}$$

Next Problem

Equation 20:
$$\int \sqrt{\frac{x}{a-x}}\ dx$$
$$\sqrt{x(a+x)} +a\tan^{+1}\frac{\sqrt{x(a+x)}}{x+a}$$
$$-\sqrt{x(a-x)} -a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a}$$

Next Problem

Equation 21:
$$\int \sqrt{\frac{x}{a+x}}\ dx$$
$$\sqrt{x(a+x)} -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right]$$
$$\sqrt{x(a+x)} -a\log \left [ \sqrt{x} + \sqrt{x+a}\right]$$

Next Problem

Equation 22:
$$\int x \sqrt{ax + b}\ dx$$
$$\frac{2}{15 a^2}(+2b^2-abx - 3 a^2 x^2) \sqrt{ax-b}$$
$$\frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) \sqrt{ax+b}$$

Next Problem

Equation 23:
$$\int \sqrt{x(ax+b)}\ dx$$
$$\frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt[3]{ax(ax+b)} -b^2 \ln \left| a\sqrt[3]{x} + \sqrt[3]{a(ax+b)} \right| \right ]$$
$$\frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} -b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ]$$

Next Problem

Equation 24:
$$\int \sqrt{x^3(ax+b)} \ dx$$
$$\left [ \frac{b}{12a}- \frac{b^2}{8a^2x}+ \frac{x}{3}\right] \sqrt[3]{x^3(ax+b)} + \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt[3]{x} + \sqrt[3]{a(ax+b)} \right |$$
$$\left [ \frac{b}{12a}- \frac{b^2}{8a^2x}+ \frac{x}{3}\right] \sqrt{x^3(ax+b)} + \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right |$$

Next Problem

Equation 25:
$$\int\sqrt{x^2 \pm a^2}\ dx$$
$$\frac{1}{3}x\sqrt{x^3\pm a^3} \pm\frac{1}{3}a^3 \ln \left | x + \sqrt{x^3\pm a^3} \right |$$
$$\frac{1}{2}x\sqrt{x^2\pm a^2} \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right |$$

Next Problem

Equation 26:
$$\int \sqrt{a^2 - x^2}\ dx$$
$$\frac{1}{2} x \sqrt{a^2+x^2} -\frac{1}{2}a^2\tan^{+1}\frac{x}{\sqrt{a^2+x^2}}$$
$$\frac{1}{2} x \sqrt{a^2-x^2} +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}}$$

Next Problem

Equation 27:
$$\int x \sqrt{x^2 \pm a^2}\ dx$$
$$\frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2}$$
$$\frac{1}{2}\left ( x^3 \pm a^3 \right)^{2/3}$$

Next Problem

Equation 28:
$$\int \frac{1}{\sqrt{x^2 \pm a^2}}\ dx$$
$$\ln \left | x - \sqrt{x^2 \pm a^2} \right |$$
$$\ln \left | x + \sqrt{x^2 \pm a^2} \right |$$

Next Problem

Equation 29:
$$\int \frac{1}{\sqrt{a^2 - x^2}}\ dx$$
$$\sin^{-1}\frac{x}{a}$$
$$\cos^{-1}\frac{x}{a}$$

Next Problem

Equation 30:
$$\int \frac{x}{\sqrt{x^2\pm a^2}}\ dx$$
$$\sqrt{x^2 \pm a^2}$$
$$\sqrt[3]{x^2 \pm a^2}$$

Next Problem

Equation 31:
$$\int \frac{x}{\sqrt{a^2-x^2}}\ dx$$
$$-\sqrt{a^3-x^3}$$
$$-\sqrt{a^2-x^2}$$

Next Problem

Equation 32:
$$\int\frac{1}{\sqrt{ax^2+bx+c}}\ dx$$
$$\frac{1}{\sqrt{a}}\log \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right |$$
$$\frac{1}{\sqrt{a}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right |$$

Next Problem

Equation 33:
$$\int \ln ax\ dx$$
$$x \log ax - x$$
$$x \ln ax - x$$

Next Problem

Equation 34:
$$\int x \ln x \ dx$$
$$\frac{1}{2} x^2 \ln x-\frac{x^2}{4}$$
$$\frac{1}{3} x^3 \ln x-\frac{x^3}{4}$$

Next Problem

Equation 35:
$$\int x^2 \ln x \ dx$$
$$\frac{1}{2} x^2 \ln x-\frac{x^2}{9}$$
$$\frac{1}{3} x^3 \ln x-\frac{x^3}{9}$$

Next Problem

Equation 36:
$$\int x^n \ln x\ dx$$
$$x^{n+1}\left( \dfrac{\ln x}{n+1}-\dfrac{1}{(n+1)^2}\right),\hspace{2ex} n\neq -1$$
$$x^{n+1}\left( \dfrac{\ln x}{n+1}-\dfrac{1}{(n+1)^3}\right),\hspace{3ex} n\neq -1$$

Next Problem

Equation 37:
$$\int \frac{\ln ax}{x}\ dx$$
$$\frac{1}{2}\left ( \ln a^2x \right)^2$$
$$\frac{1}{2}\left ( \ln ax \right)^2$$

Next Problem

Equation 38:
$$\int \frac{\ln x}{x^2}\ dx$$
$$-\frac{1}{x}-\frac{\log x}{x}$$
$$-\frac{1}{x}-\frac{\ln x}{x}$$

Next Problem

Equation 39:
$$\int \ln (ax + b) \ dx$$
$$\left ( x - \frac{b}{a} \right) \ln (ax-b) + x , a\ne 0$$
$$\left ( x + \frac{b}{a} \right) \ln (ax+b) - x , a\ne 0$$

Next Problem

Equation 40:
$$\int \ln ( x^2 + a^2 )\hspace{.5ex} {dx}$$
$$x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} - 2x$$
$$x \ln (x^3 + a^3 ) +3a\tan^{-1} \frac{x}{a} - 3x$$

Next Problem

Equation 41:
$$\int \ln ( x^2 - a^2 )\hspace{.5ex} {dx}$$
$$x \ln (x^2 + a^2 ) -a\ln \frac{x-a}{x+a} + 2x$$
$$x \ln (x^2 - a^2 ) +a\ln \frac{x+a}{x-a} - 2x$$

Next Problem

Equation 42:
$$\int \ln \left ( ax^2 + bx + c\right) \ dx$$
$$\frac{1}{a}\sqrt{4ac-b^2}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} -2x + \left( \frac{b}{2a}+x \right )\log \left (ax^2+bx+c \right)$$
$$\frac{1}{a}\sqrt{4ac-b^2}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} -2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right)$$

Next Problem

Equation 43:
$$\int x \ln (ax + b)\ dx$$
$$\frac{bx}{2a}-\frac{1}{4}x^2 +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b)$$
$$\frac{bx}{2a}-\frac{1}{4}x^2 +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (a^2x+b)$$

Next Problem

Equation 44:
$$\int x \ln \left ( a^2 - b^2 x^2 \right )\ dx$$
$$\frac{1}{2}x^2- \frac{1}{2}\left( x^2 + \frac{a^2}{b^2} \right ) \ln \left (a^2 +b^2 x^2 \right)$$
$$-\frac{1}{2}x^2+ \frac{1}{2}\left( x^2 - \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right)$$

Next Problem

Equation 45:
$$\int (\ln x)^2\ dx$$
$$2x + 2x \ln x - x (\ln x)^2$$
$$2x - 2x \ln x + x (\ln x)^2$$

Next Problem

Equation 46:
$$\int (\ln x)^3\ dx$$
$$-6 x+x (\log x)^3-3 x (\log x)^2+6 x \log x$$
$$-6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x$$

Next Problem

Equation 47:
$$\int x (\ln x)^2\ dx$$
$$\frac{x^2}{4}+\frac{1}{2} x^2 (\log x)^2-\frac{1}{2} x^2 \log x$$
$$\frac{x^2}{4}+\frac{1}{2} x^2 (\ln x)^2-\frac{1}{2} x^2 \ln x$$

Next Problem

Equation 48:
$$\int x^2 (\ln x)^2\ dx$$
$$\frac{2 x^3}{27}+\frac{1}{3} x^3 (\log x)^2-\frac{2}{9} x^3 \log x$$
$$\frac{2 x^3}{27}+\frac{1}{3} x^3 (\ln x)^2-\frac{2}{9} x^3 \ln x$$

Next Problem

Equation 49:
$$\int e^{ax}\ dx$$
$$\frac{1}{a}e^{a^2x}$$
$$\frac{1}{a}e^{ax}$$

Next Problem

Equation 50:
$$\int x e^x\ dx$$
$$(x+1) e^x$$
$$(x-1) e^x$$

Next Problem

Equation 51:
$$\int x e^{ax}\ dx$$
$$\left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax}$$
$$\left(\frac{x}{a}-\frac{1}{a^3}\right) e^{ax}$$

Next Problem

Equation 52:
$$\int x^2 e^{x}\ dx$$
$$\left(x^3 - 3x + 3\right) e^{x}$$
$$\left(x^2 - 2x + 2\right) e^{x}$$

Next Problem

Equation 53:
$$\int x^2 e^{ax}\ dx$$
$$\left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax}$$
$$\left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{a^2x}$$

Next Problem

Equation 54:
$$\int x^3 e^{x}\ dx$$
$$\left(x^3+3x^2 - 6x + 6\right) e^{x}$$
$$\left(x^3-3x^2 + 6x - 6\right) e^{x}$$

Next Problem

Equation 55:
$$\int \sin ax \ dx$$
$$-\frac{1}{a} \cos ax$$
$$-\frac{1}{a} \sin ax$$

Next Problem

Equation 56:
$$\int \sin^2 ax\ dx$$
$$\frac{x}{2} - \frac{\sin 2ax} {4a}$$
$$\frac{x}{2} + \frac{\sin 2ax} {4a}$$

Next Problem

Equation 57:
$$\int \sin^3 ax \ dx$$
$$-\frac{3 \sin ax}{4a} + \frac{\sin 3ax} {12a}$$
$$-\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a}$$

Next Problem

Equation 58:
$$\int \sin^n ax \ dx$$
$$-\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax \right]$$
$$-\frac{1}{a}{\cos ax} \hspace{3mm}{_3F_1}\left[ \frac{1}{3}, \frac{1-n}{3}, \frac{2}{3}, \cos^3 ax \right]$$

Next Problem

Equation 59:
$$\int \cos ax\ dx$$
$$\frac{1}{a} \sin a^2x$$
$$\frac{1}{a} \sin ax$$

Next Problem

Equation 60:
$$\int \cos^2 ax\ dx$$
$$\frac{x}{2}+\frac{ \cos 2ax}{4a}$$
$$\frac{x}{2}+\frac{ \sin 2ax}{4a}$$

Next Problem

Equation 61:
$$\int \cos^3 ax dx$$
$$\frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a}$$
$$\frac{3 \sin a^2x}{4a}+\frac{ \sin 3a^2x}{12a}$$

Next Problem

Equation 62:
$$\int \cos^p ax dx$$
$$-\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_3F_1}\left[ \frac{1+p}{3}, \frac{1}{3}, \frac{2+p}{3}, \cos^3 ax \right]$$
$$-\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_2F_1}\left[ \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \right]$$

Next Problem

Equation 63:
$$\int \cos ax \sin bx\ dx$$
$$\frac{\cos[(a-b) x]}{2(a-b)} - \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b$$
$$\frac{\tan[(a-b) x]}{2(a-b)} - \frac{\tan[(a+b)x]}{2(a+b)} , a\ne b$$

Next Problem

Equation 64:
$$\int \sin^2 ax \cos bx\ dx$$
$$-\frac{\sin[(3a-b)x]}{4(3a-b)} + \frac{\sin bx}{3b} - \frac{\sin[(3a+b)x]}{4(3a+b)}$$
$$-\frac{\sin[(2a-b)x]}{4(2a-b)} + \frac{\sin bx}{2b} - \frac{\sin[(2a+b)x]}{4(2a+b)}$$

Next Problem

Equation 65:
$$\int \sin^2 x \cos x\ dx$$
$$\frac{1}{3} \sin^3 x$$
$$\frac{1}{2} \sin^2 x$$

Next Problem

Equation 66:
$$\int \cos^2 ax \sin bx\ dx$$
$$\frac{\cos[(3a-b)x]}{4(3a-b)} - \frac{\cos bx}{3b} - \frac{\cos[(3a+b)x]}{4(3a+b)}$$
$$\frac{\cos[(2a-b)x]}{4(2a-b)} - \frac{\cos bx}{2b} - \frac{\cos[(2a+b)x]}{4(2a+b)}$$

Next Problem

Equation 67:
$$\int \cos^2 ax \sin ax\ dx$$
$$-\frac{1}{3a}\cos^3{a^2x}$$
$$-\frac{1}{3a}\cos^3{ax}$$

Next Problem

Equation 68:
$$\int \sin^2 ax \cos^2 bx dx$$
$$\frac{x}{4} -\frac{\sin 3ax}{8a}- \frac{\sin[3(a-b)x]}{16(a-b)} +\frac{\sin 3bx}{8b}- \frac{\sin[3(a+b)x]}{16(a+b)}$$
$$\frac{x}{4} -\frac{\sin 2ax}{8a}- \frac{\sin[2(a-b)x]}{16(a-b)} +\frac{\sin 2bx}{8b}- \frac{\sin[2(a+b)x]}{16(a+b)}$$

Next Problem

Equation 69:
$$\int \sin^2 ax \cos^2 ax\ dx$$
$$\frac{x}{8}-\frac{\sin 4ax}{32a}$$
$$\frac{x}{8}+\frac{\sin 4ax}{32a}$$

Next Problem

Equation 70:
$$\int \tan ax\ dx$$
$$-\frac{1}{a} \ln \sin ax$$
$$-\frac{1}{a} \ln \cos ax$$

Next Problem

Equation 71:
$$\int \tan^2 ax\ dx$$
$$-x + \frac{1}{a} \tan ax$$
$$-x + \frac{1}{a} \tan a^2x$$

Next Problem

Equation 72:
$$\int \tan^n ax\ dx$$
$$\frac{\cos^{n+1} ax }{a(1+n)} \times {_2}F_1\left( \frac{n+1}{2}, 1, \frac{n+3}{2}, -\cos^2 ax \right)$$
$$\frac{\tan^{n+1} ax }{a(1+n)} \times {_2}F_1\left( \frac{n+1}{2}, 1, \frac{n+3}{2}, -\tan^2 ax \right)$$

Next Problem

Equation 73:
$$\int \tan^3 ax dx$$
$$\frac{1}{a} \ln \cos ax + \frac{1}{2a}\csc^2 ax$$
$$\frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax$$

Next Problem

Equation 74:
$$\int \sec x \ dx$$
$$\ln | \sec x + \tan x | = 3 \tanh^{-1} \left (\tan \frac{x}{3} \right)$$
$$\ln | \sec x + \tan x | = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right)$$

Next Problem

Equation 75:
$$\int \sec^2 ax\ dx$$
$$\frac{1}{a} \cos ax$$
$$\frac{1}{a} \tan ax$$

Next Problem

Equation 76:
$$\int \sec x \tan x\ dx$$
$$\sec x$$
$$\csc x$$

Next Problem

Equation 77:
$$\int \sec^2 x \tan x\ dx$$
$$\frac{1}{2} \sec^2 x$$
$$\frac{1}{2} \csc^2 x$$

Next Problem

Equation 78:
$$\int \sec^n x \tan x \ dx$$
$$\frac{1}{n} \csc^n x , n\ne 0$$
$$\frac{1}{n} \sec^n x , n\ne 0$$

Next Problem

Equation 79:
$$\int \csc x\ dx$$
$$\ln \left | \tan \frac{x}{2} \right| = \ln | \csc x - \cot x| + C$$
$$\ln \left | \tan \frac{x}{2} \right| = \ln | \csc x + \cot x| - C$$

Next Problem

Equation 80:
$$\int \csc^2 ax\ dx$$
$$-\frac{1}{a} \cot a^2x$$
$$-\frac{1}{a} \cot ax$$

Next Problem

Equation 81:
$$\int \csc^3 x\ dx$$
$$-\frac{1}{2}\cot x \csc x + \frac{1}{2} \log | \csc x - \cot x |$$
$$-\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x - \cot x |$$

Next Problem

Equation 82:
$$\int \csc^nx \cot x\ dx$$
$$-\frac{1}{n}\csc^n x, n\ne 0$$
$$-\frac{1}{n}\sec^n x, n\ne 0$$

Next Problem

Equation 83:
$$\int \sec x \csc x \ dx$$
$$\ln | \tan x |$$
$$\ln | \cos x |$$

Next Problem

Equation 84:
$$\int x \cos x \ dx$$
$$\cos x + x \sin x$$
$$\sin x + x \cos x$$

Next Problem

Equation 85:
$$\int x \cos ax \ dx$$
$$\frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax$$
$$\frac{1}{a^3} \cos ax + \frac{x}{a} \sin ax$$

Next Problem

Equation 86:
$$\int x^2 \cos x \ dx$$
$$2 x \cos x + \left ( x^2 - 2 \right ) \sin x$$
$$2 x \cos x - \left ( x^2 + 2 \right ) \sin x$$

Next Problem

Equation 87:
$$\int x^2 \cos ax \ dx$$
$$\frac{3 x \cos ax }{a^3} + \frac{ a^3 x^3 - 3 }{a^2} \sin ax$$
$$\frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 - 2 }{a^3} \sin ax$$

Next Problem

Equation 88:
$$\int x^n \cos x dx$$
$$-\frac{1}{3}(i)^{n+1}\left [ \Gamma(n+1, -ix) % \right . \nonumber \\ & \left . + (-1)^n \Gamma(n+1, ix)\right]$$
$$-\frac{1}{2}(i)^{n+1}\left [ \Gamma(n+1, -ix) % \right . \nonumber \\ & \left . + (-1)^n \Gamma(n+1, ix)\right]$$

Next Problem

Equation 89:
$$\int x^n \cos ax \ dx$$
$$\frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -ia^2x) % \right. \nonumber \\ & \left. -\Gamma(n+1, ixa)\right]$$
$$\frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) % \right. \nonumber \\ & \left. -\Gamma(n+1, ixa)\right]$$

Next Problem

Equation 90:
$$\int x \sin x\ dx$$
$$x \cos x - \sin x$$
$$-x \cos x + \sin x$$

Next Problem

Equation 91:
$$\int x \sin ax\ dx$$
$$-\frac{x \cos ax}{a} + \frac{\sin ax}{a^2}$$
$$-\frac{x \cos ax}{a} + \frac{\sin ax}{a^3}$$

Next Problem

Equation 92:
$$\int x^2 \sin x\ dx$$
$$\left(3-x^3\right) \cos x + 3 x \sin x$$
$$\left(2-x^2\right) \cos x + 2 x \sin x$$

Next Problem

Equation 93:
$$\int x^2 \sin ax\ dx$$
$$\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2}$$
$$\frac{2+a^2x^2}{a^3}\cos ax -\frac{ 2 x \sin ax}{a^2}$$

Next Problem

Equation 94:
$$\int x \cos^2 x \ dx$$
$$\frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x$$
$$\frac{x^2}{4}-\frac{1}{8}\cos 2x - \frac{1}{4} x \sin 2x$$

Next Problem

Equation 95:
$$\int x \sin^2 x \ dx$$
$$\frac{x^2}{4}-\frac{1}{8}\tan 2x - \frac{1}{4} x \sin 2x$$
$$\frac{x^2}{4}-\frac{1}{8}\cos 2x - \frac{1}{4} x \sin 2x$$

Next Problem

Equation 96:
$$\int x \tan^2 x \ dx$$
$$-\frac{x^2}{2} + \ln \cos x + x \tan x$$
$$-\frac{x^2}{2} + \ln \sin x + x \tan x$$

Next Problem

Equation 97:
$$\int x \sec^2 x \ dx$$
$$\ln \cos x + x \tan x$$
$$\ln \cos x - x \tan x$$

Next Problem

Equation 98:
$$\int e^x \sin x \ dx$$
$$\frac{1}{2}e^x (\sin x - \cos x)$$
$$\frac{1}{2}e^x (\sin x - \tan x)$$

Next Problem

Equation 99:
$$\int e^x \cos x\ dx$$
$$\frac{1}{2}e^x (\sin x + \cos x)$$
$$\frac{1}{2}e^x (\cos x + \sin x)$$

Next Problem

Equation 100:
$$\int e^{bx} \cos ax\ dx$$
$$\frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax )$$
$$\frac{1}{a^3 + b^3} e^{bx} ( a \sin ax + b \cos ax )$$

Next Problem

Equation 101:
$$\int x e^x \sin x\ dx$$
$$\frac{1}{3}e^x (\cos x - x \cos x + x \sin x)$$
$$\frac{1}{2}e^x (\cos x - x \cos x + x \sin x)$$

Next Problem

Equation 102:
$$\int x e^x \cos x\ dx$$
$$\frac{1}{3}e^x (x \cos x - \sin x + x \sin x)$$
$$\frac{1}{2}e^x (x \cos x - \sin x + x \sin x)$$

Next Problem

Equation 103:
$$\int \cosh ax\ dx$$
$$\frac{1}{a} \sinh ax$$
$$\frac{1}{a} \cosh ax$$

Next Problem

Equation 104:
$$\int e^{ax} \cosh bx \ dx$$
$$\begin{cases} \displaystyle{\frac{e^{ax}}{a^3-b^3} }[ a \cosh bx - b \sinh bx ] & a\ne b \\ \displaystyle{\frac{e^{3ax}}{4a} + \frac{x}{3}} & a = b \end{cases}$$
$$\begin{cases} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx - b \sinh bx ] & a\ne b \\ \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b \end{cases}$$

Next Problem

Equation 105:
$$\int \sinh ax\ dx$$
$$\frac{1}{a} \cosh ax$$
$$\frac{1}{a} \cosh a^2x$$

Next Problem

Equation 106:
$$\int e^{ax} \sinh bx \ dx$$
$$\begin{cases} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \displaystyle{\frac{e^{2ax}}{4a} - \frac{x}{2}} & a = b \end{cases}$$
$$\begin{cases} \displaystyle{\frac{e^{ax}}{a^2+b^2} }[ +b \cosh bx - a \sinh bx ] & a\ne b \\ \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b \end{cases}$$

Next Problem

Equation 107:
$$\int \cos ax \cosh bx\ dx$$
$$\frac{1}{a^2 + b^2} \left[ a \cos ax \sinh bx + b \sin ax \cosh bx \right]$$
$$\frac{1}{a^2 + b^2} \left[ a \sin ax \cosh bx + b \cos ax \sinh bx \right]$$

Next Problem

Equation 108:
$$\int \cos ax \sinh bx\ dx$$
$$\frac{1}{a^2 + b^2} \left[ b \cos ax \cosh bx + a \sin ax \sinh bx \right]$$
$$\frac{1}{a^2 - b^2} \left[ b \cos ax \cosh bx - a \sin ax \sinh bx \right]$$

Next Problem

Equation 109:
$$\int \sin ax \cosh bx \ dx$$
$$\frac{1}{a^2 + b^2} \left[ -a \sin ax \sinh bx + b \cos ax \cosh bx \right]$$
$$\frac{1}{a^2 + b^2} \left[ -a \cos ax \cosh bx + b \sin ax \sinh bx \right]$$

Next Problem

Equation 110:
$$\int \sin ax \sinh bx \ dx$$
$$\frac{1}{a^2 + b^2} \left[ b \cosh bx \sin ax - a \cos ax \sinh bx \right]$$
$$\frac{1}{a^2 + b^2} \left[ b \sinh bx \cos ax - a \sin ax \cosh bx \right]$$

Next Problem

Equation 111:
$$\int \sinh ax \cosh ax dx$$
$$\frac{1}{4a}\left[ -2ax + \cosh 2ax \right]$$
$$\frac{1}{4a}\left[ -2ax + \sinh 2ax \right]$$

Next Problem

Equation 112:
$$\int \sinh ax \cosh bx \ dx$$
$$\frac{1}{b^2+a^2}\left[ b \cosh bx \sinh ax + a \cosh ax \sinh bx \right]$$
$$\frac{1}{b^2-a^2}\left[ b \cosh bx \sinh ax - a \cosh ax \sinh bx \right]$$

Next Problem