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INTEGRALS > Simple functions

Integrals of simple functions

Rational functions

$\int \,dx = x + C$

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ if }n \ne -1$

$\int \frac{1}{x}\,dx = \ln{\left|x\right|} + C$

$\int \frac{1}{1+x^2} \, dx = \arctan{x} + C$

Logarithms

$\int \ln {x}\,dx = x \ln {x} - x + C$

$\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C$

Exponential functions

$\int e^x\,dx = e^x + C$

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

Irrational functions

$\int {1 \over \sqrt{1-x^2}} \, dx = \arcsin {x} + C$

$\int {-1 \over \sqrt{1-x^2}} \, dx = \arccos {x} + C$

$\int {x \over \sqrt{x^2-1}} \, dx = \mbox{arcsec}\,{x} + C$

Trigonometric functions

$\int \cos{x}\, dx = \sin{x} + C$

$\int \sin{x}\, dx = -\cos{x} + C$

$\int \tan{x} \, dx = -\ln{\left| \cos {x} \right|} + C_1 = \ln{\left| \sec {x} \right|} + C_2$

$\int \csc{x} \, dx = \ln{\left| \csc{x} - \cot{x}\right|} + C$

$\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C$

$\int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C$

$\int \sec^2 x \, dx = \tan x + C$

$\int \csc^2 x \, dx = -\cot x + C$

$\int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + C$

$\int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C$

$\int \csc{x} \, \cot{x} \, dx = - \csc{x} + C$

$\int \arctan{x} \, dx = x \, \arctan{x} + \ln \sqrt{1 + x^2} + C$

Hyperbolic functions

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

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

$\int \tanh x \, dx = \ln (\cosh x) + C$

$\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C$

$\int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C$

$\int \coth x \, dx = \ln|\sinh x| + C$

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