PlanetMath: table of integrals

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table of integrals

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Below are some tables of some real-valued functions and their corresponding indefinite integrals.

Polynomials and powers

	$\displaystyle{\int f(x)\, dx}$
for $n\ne -1$	$\displaystyle{\frac{x^{n+1}}{n\!+\!1}}+C$
$x^{-1}$	$\ln\vert x\vert+C$
$\vert x\vert^n$ for $n\ne -1$	$\displaystyle\frac{x\vert x\vert^n}{n\!+\!1}+C$
$\vert x\vert^{-1}$	$\displaystyle\frac{x\ln\vert x\vert}{\vert x\vert}+C$

Exponential and logarithmic functions

	$\displaystyle{\int f(x)\, dx}$

$e^{kx}$ for $k\neq 0$	$\displaystyle\frac{e^{kx}}{k}+C$
for	$\displaystyle\frac{a^x}{\ln{a}}+C$
$\ln{x}$	$x\ln{x}-x+C$
$(\ln{x})^2$	$x[(\ln{x})^2-2\ln{x}+2]+C$
$\displaystyle\frac{1}{\ln{x}}$	$\operatorname{Li}{x}+C$
$\ln(\ln{x})$	$x\ln\ln{x}-\operatorname{Li}{x}+C$

Trigonometric functions

	$\displaystyle{\int f(x)\, dx}$
$\cos{x}$	$\sin{x}+C$
$\sin{x}$	$-\cos{x}+C$
$\cot{x}$	$\ln\vert\sin{x}\vert+C$
$\tan{x}$	$-\ln\vert\cos{x}\vert+C$
$\sec{x}$	$\ln\vert\sec{x}+\tan{x}\vert+C$
$\csc{x}$	$-\ln\vert\csc{x}+\cot{x}\vert+C$
$\sec^2{x}$	$\tan{x}+C$
$\csc^2{x}$	$-\cot{x}+C$
$\sec{x}\tan{x}$	$\sec{x}+C$
$\csc{x}\cot{x}$	$-\csc{x}+C$
$\displaystyle\frac{1}{1+x^2}$	$\arctan{x}+C$
$\displaystyle\frac{1}{\sqrt{1-x^2}}$	$\arcsin{x}+C$

Hyperbolic functions

	$\displaystyle{\int f(x)\, dx}$
$\cosh{x}$	$\sinh{x}+C$
$\sinh{x}$	$\cosh{x}+C$
$\tanh{x}$	$\ln(\cosh{x})+C$
$\coth{x}$	$\ln\vert\sinh{x}\vert+C$
$\operatorname{sech}^2{x}$	$\tanh{x}+C$
$\operatorname{csch}^2{x}$	$-\coth{x}+C$
$\operatorname{sech}{x}\tanh{x}$	$-\operatorname{sech}{x}+C$
$\operatorname{csch}{x}\coth{x}$	$-\operatorname{csch}{x}+C$

Cyclometric functions

	$\displaystyle{\int f(x)\, dx}$
$\arccos{x}$	$x\arccos{x}-\sqrt{1-x^2}+C$
$\arcsin{x}$	$x\arcsin{x}+\sqrt{1-x^2}+C$
$\operatorname{arccot}{x}$	$x\operatorname{arccot}{x}+\ln\sqrt{1+x^2}+C$
$\arctan{x}$	$x\arctan{x}-\ln\sqrt{1+x^2}+C$
$\operatorname{arcsec}{x}$	$x\operatorname{arcsec}{x}-\ln(x+\sqrt{x^2-1})+C$

Some square roots

	$\displaystyle{\int f(x)\, dx}$
$\sqrt{x}$	$\frac{2}{3}x\sqrt{x}+C$
$\sqrt{x^2+1}$	$\displaystyle\frac{x}{2}\sqrt{x^2+1}+\frac{1}{2}\operatorname{arsinh}{x}+C$
$\sqrt{x^2-1}$	$\displaystyle\frac{x}{2}\sqrt{x^2-1}-\frac{1}{2}\operatorname{arcosh}{x}+C$
$\displaystyle\frac{1}{\sqrt{x^2+1}}$	$\operatorname{arsinh}{x}+C$
$\displaystyle\frac{1}{\sqrt{x^2-1}}$	$\operatorname{arcosh}{x}+C$

Remark 1

above denotes an arbitrary constant real number; $\operatorname{Li}$ is the logarithmic integral.

Remark 2 Note that the table can only be used to compute a definite integral when the integrand is continuous on the domain of integration. For example, note the following erroneous calculation:

$\displaystyle \int\limits_{-1}^1 \vert x\vert^{-1} \, dx=\frac{x\ln\vert x\vert... ...c{1\ln\vert 1\vert}{\vert 1\vert}-\frac{-1\ln\vert-1\vert}{\vert-1\vert}=0-0=0$

The above calculation is incorrect since $\vert x\vert^{-1}$ is not continuous at .

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Cross-references: integral, continuous at, domain, continuous, definite integral, logarithmic integral, real number, indefinite integrals, functions

This is version 29 of table of integrals, born on 2007-10-12, modified 2008-04-01.
Object id is 9991, canonical name is IntegralTables.
Accessed 1307 times total.

Classification:

AMS MSC:

26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)

Pending Errata and Addenda

1. Cross-linking and transforms by rspuzio on 2008-05-13 02:12:54
[ View all 1 ]

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