Table of Indefinite Integrals

This is a table of common antiderivatives and definite integrals for the Calculus students. Remember after each indefinite integral there is a general constant of integration (+C).

Antiderivatives

Elementary

$\int x^{n}\,dx={\frac {1}{n+1}}x^{n+1},\,n\neq -1$

$\int {\frac {1}{x}}\,dx=\ln |x|$

$\int e^{x}\,dx=e^{x}$

$\int \sin x\,dx=-\cos x$

$\int \cos x\,dx=\sin x$

$\int \tan x\,dx=-\ln \cos x$

$\int \sinh x\,dx=\cosh x$

$\int \cosh x\,dx=\sinh x$

Commonly encountered

$\int \ln x\,dx=x\ln x-x$

$\int a^{x}\,dx={\frac {1}{\ln a}}a^{x}$

$\int {\frac {1}{x^{2}+a^{2}}}\,dx={\frac {1}{a}}\tan ^{-1}{\frac {x}{a}}={\frac {1}{a}}\arctan {\frac {x}{a}}$

$\int {\frac {1}{a^{2}-x^{2}}}\,dx={\frac {1}{2}}\ln \left|{\frac {x+a}{x-a}}\right|$

$\int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\,dx=\sin ^{-1}{\frac {x}{a}}=\arcsin {\frac {x}{a}}$

$\int {\frac {1}{\sqrt {x^{2}-a^{2}}}}\,dx=\cosh ^{-1}{\frac {x}{a}}=\ln(x+{\sqrt {x^{2}-a^{2}}})$

$\int {\frac {1}{\sqrt {x^{2}+a^{2}}}}\,dx=\sinh ^{-1}{\frac {x}{a}}=\ln(x+{\sqrt {x^{2}+a^{2}}})$

See also

• Integration
• Methods of integration

External links

• Integral table