including integral

C is a constant of integration.
----------
$\intx^ndx =\frac{x^{n+1}}{n+1}+C$
n is not equal to -1.
----------
$\int\frac{1}{x}dx=\log|x|+C$
----------
$\int\sinxdx =-\cosx+C$
----------
$\int\cosxdx=\sin x+C$
----------
$\int\tan{x}dx=-\log(\cos{x})+C$
----------
$\int\sin^2{x}dx=\frac{1}{2}(x-\sin{x}\cos{x})+C$
----------
$\int\cos^2{x}dx=\frac{1}{2}(x+\sin{x}\cos{x})+C$
----------
$\int\tan^2{x}dx=\tan{x}-x+C$
----------
$\int\tan^2{x}dx=\tan{x}-x+C$
----------
$\int\frac{1}{\cos{x}}dx=\log(\tan{x}+\frac{1}{\cos{x}})+C$
----------
$\int\frac{1}{\tan{x}}dx=\log(\sin{x})+C$
----------
$\int\frac{1}{1+\sin{x}}dx=\frac{2\sin(\frac{x}{2})}{\sin(\frac{x}{2})+\cos(\frac{x}{2})}+C$
----------
$\int\frac{1}{1+\cos{x}}dx=\tan(\frac{x}{2})+C$
----------
$\int\frac{1}{1+\tan{x}}dx=\frac{1}{2}(x+\log(\sin{x}+\cos{x}))+C$
----------
$\int \frac{1}{1+\sin^2{x}} dx =\frac{1}{2}(3x-\sin{x}\cos{x})+C$
----------
$\int \frac{1}{1+\cos^2{x}} dx =\frac{1}{\sqrt{2}}\tan^{-1}{(\frac{\tan{x}}{\sqrt{2}})}+C$
----------
$\int\frac{1}{1+\tan^2{x}} dx =\frac{1}{2}(x+\sin{x}\cos{x})+C$
----------
$\int \cosh x dx = \sinh x + C$
----------
$\int\sinhxdx=\coshx+C$
----------
$\int\tanh{x}dx=\log(\cosh{x})+C$
----------
$\int\frac{1}{\sinh{x}}dx=\log(\tanh{(\frac{x}{2}))+C$
----------
$\int\frac{1}{\cosh{x}}dx=2\tan^{-1}(\tanh{(\frac{x}{2}))+C$
----------
$\int\frac{1}{\tanh{x}}dx=\log(\sinh{x})+C$
----------
\int\frac{1}{\cosh^2x}dx=\tanhx+C
----------
$\int \frac{1}{\sqrt{1-x^2}dx=\arcsin +C~~(-1
----------
$\int-\frac{1}{\sqrt{1-x^2}}dx=\arccosx+C~~(-1
----------
$\int\frac{1}{1+x^2} dx=\arctanx+ C$
----------
$\int\frac{1}{1+x^3} dx=\frac{1}{6}\log \frac{(x+1)^2}{x^2-x+1}+\frac{1}{\sqrt{3}}\arctan (\frac{2x-1}{\sqrt{3}})+C
----------
$\int\frac{1}{1+x^4}dx=\frac{1}{4\sqrt{2}}\log (\frac{x^2+\sqrt{2}x+1}{x^2-\sqrt{2}x+1})+\frac{1}{2\sqrt{2}}\{\arctan ({\sqrt{2}x+1})+\arctan ({\sqrt{2}x-1})\}+C$
----------
$\int\frac{x}{1+x^4}dx=\frac{1}{4}\log{(1+x^4)}$
----------
$\int\frac{x^2}{1+x^4}dx=\frac{1}{4\sqrt{2}}\log (\frac{x^2+\sqrt{2}x+1}{x^2-\sqrt{2}x+1})+\frac{1}{2\sqrt{2}}\arctan{\frac{\sqrt{2}x}{1-x^2}+C$
----------
$\int\frac{x^3}{1+x^4}dx=\frac{1}{2}\arctan{x^2}$
----------


$\int\frac{1}{1-x^2}dx=\frac{1}{2}\log{\frac{1+x}{1-x}}+C$

----------


$\int\frac{1}{1-x^3}dx=\frac{1}{3}\log{\frac{\sqrt{1+x+x^2}}{1-x}}+\frac{1}{\sqrt{3}}\arctan{\frac{\sqrt{3}x}{2+x}}+C$

----------


$\int\frac{1}{1-x^4}dx=\frac{1}{4}\log{\frac{1+x}{1-x}}+\frac{1}{2}\arctan{x}+C$

----------

$\int e^x dx=e^x+C$
----------
$\int a^x dx=\frac{a^x}{\log{a}}+C$
----------
$\int \frac{1}{1+e^x} dx=x-\log{(1+e^x)}+C$
----------
$\int \log{x}dx=x(\log{x}-1)+C$
----------
$\int x\log{x}dx=\frac{1}{4}x^2(2\log{x}-1)+C$
----------
$\int\log^2{x}dx=x(\log^2{x}-2\log{x}+2)+C$
----------
$\int \frac{\log{x}}{x}dx=\frac{1}{2}\log^2{x}+C$
----------
$\int_{-\infty}^{\infty} \frac{1}{(1+x^2)^{n+1}} dx = \frac{\pi (2n)!}{2^{2n}(n!)^2} $
----------
$ \int_{0}^{\infty} e^{-a^2x^2}dx = \frac{\sqrt{\pi}}{2|a|} $
----------
$\int_{-\infty}^{\infty} \sin (x^2) dx =\sqrt{\frac{\pi}{2}}$
----------
$\int_{0}^{\infty} \frac{x}{e^x-e^{-x}}dx=\frac{\pi^2}{8}$
----------
$\int_{0}^{\infty}\frac{x}{e^x+1}dx=\frac{\pi^2}{12}$
----------
$\int_{0}^{\infty}\frac{x}{e^x-1}dx=\frac{\pi^2}{12}$
----------
$\int_{0}^{\infty}\log(\frac{e^x-1}{e^x+1})dx=\frac{\pi^2}{4}$
----------
$\int_{0}^{1} \frac{\log{(1+x^{4+\sqrt{15}})}}{1+x}dx=\frac{\pi^2}{12}(2-\sqrt{15})+\log(\frac{1+\sqrt{5}}{2})\log(2+\sqrt{3})+\log2\log(\sqrt{3}+\sqrt{5})$
----------
$\int_{0}^{1}\frac{x^{n-1}}{1+x^n} \log\log{(\frac{1}{x})}dx =-\frac{\log(2)\log(2n^2)}{2n}$
----------
$\int_{0}^{1}\frac{1-x}{\log{x}} \sum_{k=1}^{\infty}x^{2^{k}}dx = -\log2$
----------
$\int_{0}^{\infty}\frac{\sin x}{x}dx=\frac{\pi}{2}$
----------
$\int_{0}^{\infty} \frac{\sin^2{x}}{x^2}dx=\frac{\pi}{2}$
----------
$\int_{0}^{\infty} \frac{\sin^2{x}}{x^2}dx=\frac{\pi}{2}$
----------
$\int_{0}^{\infty} \frac{\sin^2{x}}{x^2}dx=\frac{\pi}{2}$
----------
\int_{0}^{\infty} \frac{\sin^6{x}}{x^6}=\frac{11\pi}{40}
----------
\int_{0}^{\infty} \frac{\sin^8{x}}{x^8}=\frac{151\pi}{630}
----------
\int_{0}^{\infty} \frac{\sin^{10}{x}}{x^{10}}=\frac{15619\pi}{72576}
----------
$\int_{0}^{\infty} \frac{1}{1+\tan^m{x}}dx=\frac{\pi}{4}$(m is an integer.)
----------
$\int_{0}^{\frac{\pi}{2}}\frac{1}{a^2 \cos^2{x}+b^2\sin^2{x}}}dx=\frac{\pi}{2ab}~~(ab \neq 0)$
----------
$\int_{0}^{\infty} e^{-ax}\cos{(bx)}dx =\frac{a}{a^2+b^2}$
----------
$\int_{0}^{\infty} e^{-ax}\sin{(bx)}dx =\frac{b}{a^2+b^2}$
----------
$\int_{0}^{\infty} e^{-tx^2} \cos{x^2}dx=\sqrt{\frac{\pi}{8}}\sqrt{\frac{\sqrt{1+t^2}+t}{1+t^2}}$
----------
$\int_{0}^{\infty} e^{-tx^2} \sin{x^2}dx=\sqrt{\frac{\pi}{8}}\sqrt{\frac{\sqrt{1+t^2}-t}{1+t^2}}$
----------
$\int_{0}^{2\pi}e^{\cos{x}}\cos{(\sin{x})dx=2\pi$
----------
$\int_{0}^{1} \frac{\log^2{x}}{x} \log{(1+x)}dx =\frac{7\pi^4}{360}$
----------
$\int_{0}^{1} \frac{\log^2{x}}{x} \log{(1-x)}dx =-\frac{\pi^4}{45}$
----------
$\int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx =-\frac{\pi^4}{15}$
----------
$\int_{-\pi}^{\pi} \log(2\cos{\frac{x}{2}}) dx =0$
----------
$\int_{0}^{\pi} \log(\sin{x}) dx =-\pi\log{2}$
----------
$\int_0^{\pi}(\log(\sin{x}))^2dx=\frac{\pi ^3}{12}+\pi\log ^2(2)$
----------
$\int_0^\infty \frac{e^{ax}-e^{-ax}}{e^{\pi x}-e^{-\pi x}} \frac{dx}{1+n^2 x^2} \quad = \quad \sum_{k=1}^\infty (-1)^{k+1} \frac{\sin(ka)}{1+kn}$
----------
$\int_{0}^{1}\frac{1}{1+x}\sum_{k=1}^{\infty}x^{2^k}dx=1-\lim_{n\rightarrow\infty}(\sum_{k=1}^{n}\frac{1}{k}-\log{n})$
----------
$\int_{0}^{1}\frac{1+2x}{1+x+x^2}\sum_{k=1}^{\infty}x^{3^k}dx=1-\lim_{n\rightarrow\infty}(\sum_{k=1}^{n}\frac{1}{k}-\log{n})$
----------
 $\int_{0}^{\infty}e^{-3\pi x^{2}}\frac{\sinh \pi x}{\sinh 3\pi x}\,dx = \frac{1}{e^{2\pi/3}\sqrt{3}}\sum_{n = 0}^{\infty}\frac{e^{-2n(n + 1)\pi}}
{\prod_{m=0}^{n}(1 + e^{-(2m+1)\pi})^{2}}$
----------
$\int_{0}^{1} \int_{0}^{1} \frac{1}{1-xy} dxdy= \frac{\pi^2}{6}$
----------
$\int_{0}^{1} \int_{0}^{1} \frac{1}{2-xy} dxdy= \frac{\pi^2}{12}-\frac{(\log{2})^2}{2}$
----------
$\int_{0}^{1} \int_{0}^{1} \frac{y}{1-x^3y^3} dxdy= \frac{\pi}{3\sqrt{3}}$

----------