Integration Table

\(\displaystyle\int a\,dx=ax+C\)
\(\displaystyle\int x^a\,dx=\frac{x^{a+1}}{a+1}+C\,\,,a\neq -1\)
\(\displaystyle\int\frac{1}{x}\,dx=\ln \vert x\vert+C\)
\(\displaystyle\int e^x\,dx=e^x+C\)
\(\displaystyle\int a^x\,dx=\frac{a^x}{\ln a}+C\)
\(\displaystyle\int\sin x\,dx=-\cos x+C\)
\(\displaystyle\int\cos x\,dx=\sin x+C\)
\(\displaystyle\int\sec^2x\,dx=\tan x+C\)
\(\displaystyle\int\csc^2x\,dx=-\cot x+C\)
\(\displaystyle\int\sec x\tan x\,dx=\sec x+C\)
\(\displaystyle\int\csc x\cot x\,dx=-\csc x+C\)
\(\displaystyle\int\sec x\,dx=\ln\vert\sec x+\tan x\vert+C\)
\(\displaystyle\int\csc x\,dx=-\ln\vert\csc x+\cot x\vert+C\)
\(\displaystyle\int\tan x\,dx=-\ln\vert\cos x\vert +C\)
\(\displaystyle\int\cot x\,dx=\ln\vert\sin x\vert +C\)
\(\displaystyle\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\left(\frac{x}{a}\right)+C\)
\(\displaystyle\int\frac{dx}{a^2+x^2}=\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C\)
\(\displaystyle\int\frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}\sec^{-1}\left(\frac{\vert x\vert}{a}\right)+C\,\,,a>0\)
\(\displaystyle\int\frac{dx}{x^2-a^2}=\frac{1}{2a}\ln\left\vert\frac{x-a}{x+a}\right\vert+C\)
\(\displaystyle\int\frac{dx}{\sqrt{x^2\pm a^2}}=\ln\left\vert x+\sqrt{x^2\pm a^2}\right\vert+C\)
\(\displaystyle\int \frac{f'(x)}{f(x)}\,dx=\ln\left\vert f(x)\right\vert+C\)
\(\displaystyle\int f'(x)e^{f(x)}\,dx=e^{f(x)}+C\)
\(\displaystyle\int_a^b f'(x)\,dx=f(b)-f(a)\)
\(\displaystyle\int_a^b f(x)g'(x)\,dx=f(x)g(x)\bigg\vert_a^b-\int_a^bf'(x)g(x)\,dx\)