| Function f | Antiderivatives F of f |
|---|---|
| \(f(x) =0\) | \(F(x) =C\) |
| \(f(x) =1\) | \(F(x) =x+C\) |
| \(f(x) =x^{a},\quad a\neq -1\) | \(F(x) =\dfrac{x^{a+1}}{a+1}+C\) |
| \(f(x) =x^{-1}=\dfrac{1}{x}\) | \(F(x) =\ln \vert x\vert +C\) |
| \(f(x) =e^{x}\) | \(F(x) =e^{x}+C\) |
| \(f(x) =a^{x}\) | \(F(x) =\dfrac{a^{x}}{\ln a}+C,\quad a>0, a\neq 1\) |
| \(f(x) =\sin x\) | \(F(x) =-\cos x+C\) |
| \(f(x) =\cos x\) | \(F(x) =\sin x+C\) |
| \(f(x) =\sec^{2}x\) | \(F(x) =\tan x+C\) |
| \(f(x) =\sec x\tan x\) | \(F(x) =\sec x+C\) |
| \(f(x) =\csc x\cot x\) | \(F(x) =-\csc x+C\) |
| \(f(x) =\csc ^{2}x\) | \(F(x) =-\cot x+C\) |
| \(f(x) =\dfrac{1}{\sqrt{1-x^{2}}},\quad \vert x\vert <1\) | \(F(x) =\sin ^{-1}x+C\) |
| \(f(x) =\dfrac{1}{1+x^{2}}\) | \(F(x) =\tan ^{-1}x+C\) |
| \(f(x) =\dfrac{1}{x\sqrt{x^{2}-1}},\quad \vert x\vert >1\) | \(F(x) =\sec^{-1}x+C\) |
| \(f(x) =\sinh x\) | \(F(x) =\cosh x+C\) |
| \(f(x) =\cosh x\) | \(F(x) =\sinh x+C\) |