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table of integrals
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(Feature)
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Below are some tables of some real-valued functions and their corresponding indefinite integrals.
| $f(x)$ |
$\displaystyle{\int f(x)\, dx}$ |
derivation |
| $x^n$ for $n\ne -1$ |
$\displaystyle{\frac{x^{n+1}}{n\!+\!1}}+C$ |
here |
| $x^{-1}$ |
$\ln|x|+C$ |
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| $|x|^n$ for $n\ne -1$ |
$\displaystyle\frac{x|x|^n}{n\!+\!1}+C$ |
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| $|x|^{-1}$ |
$\displaystyle\frac{x\ln|x|}{|x|}+C$ |
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| $f(x)$ |
$\displaystyle{\int f(x)\, dx}$ |
derivation |
| $e^x$ |
$e^x+C$ |
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| $e^{kx}$ for $k\neq 0$ |
$\displaystyle\frac{e^{kx}}{k}+C$ |
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| $a^x$ for $a>0$ |
$\displaystyle\frac{a^x}{\ln{a}}+C$ |
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| $\ln{x}$ |
$x\ln{x}-x+C$ |
here |
| $(\ln{x})^2$ |
$x[(\ln{x})^2-2\ln{x}+2]+C$ |
here |
| $\displaystyle\frac{1}{\ln{x}}$ |
$\Li{x}+C$ |
Li |
| $\ln(\ln{x})$ |
$x\ln\ln{x}-\Li{x}+C$ |
here |
| $f(x)$ |
$\displaystyle{\int f(x)\, dx}$ |
derivation |
| $\cos{x}$ |
$\sin{x}+C$ |
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| $\sin{x}$ |
$-\cos{x}+C$ |
here |
| $\cot{x}$ |
$\ln|\sin{x}|+C$ |
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| $\tan{x}$ |
$-\ln|\cos{x}|+C$ |
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| $\sec{x}$ |
$\ln|\sec{x}+\tan{x}|+C$ |
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| $\csc{x}$ |
$-\ln|\csc{x}+\cot{x}|+C$ |
here |
| $\displaystyle\frac{1}{\sin{x}}$ |
$\displaystyle\ln\left|\tan\frac{x}{2}\right|+C$ |
here |
| $\sec^2{x}$ |
$\tan{x}+C$ |
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| $\csc^2{x}$ |
$-\cot{x}+C$ |
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| $\sec{x}\tan{x}$ |
$\sec{x}+C$ |
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| $\csc{x}\cot{x}$ |
$-\csc{x}+C$ |
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| $\displaystyle\frac{1}{1+x^2}$ |
$\arctan{x}+C$ |
here |
| $\displaystyle\frac{1}{\sqrt{1-x^2}}$ |
$\arcsin{x}+C$ |
here |
| $f(x)$ |
$\displaystyle{\int f(x)\, dx}$ |
derivation |
| $\cosh{x}$ |
$\sinh{x}+C$ |
here |
| $\sinh{x}$ |
$\cosh{x}+C$ |
here |
| $\tanh{x}$ |
$\ln(\cosh{x})+C$ |
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| $\coth{x}$ |
$\ln|\sinh{x}|+C$ |
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| $\sech^2{x}$ |
$\tanh{x}+C$ |
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| $\csch^2{x}$ |
$-\coth{x}+C$ |
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| $\sech{x}\tanh{x}$ |
$-\sech{x}+C$ |
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| $\csch{x}\coth{x}$ |
$-\csch{x}+C$ |
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| $f(x)$ |
$\displaystyle{\int f(x)\, dx}$ |
derivation |
| $\arccos{x}$ |
$x\arccos{x}-\sqrt{1-x^2}+C$ |
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| $\arcsin{x}$ |
$x\arcsin{x}+\sqrt{1-x^2}+C$ |
here |
| $\arccot{x}$ |
$x\arccot{x}+\ln\sqrt{1+x^2}+C$ |
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| $\arctan{x}$ |
$x\arctan{x}-\ln\sqrt{1+x^2}+C$ |
here |
| $\arcsec{x}$ |
$x\arcsec{x}-\ln(x+\sqrt{x^2-1})+C$ |
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| $f(x)$ |
$\displaystyle{\int f(x)\, dx}$ |
derivation |
| $\sqrt{x}$ |
$\frac{2}{3}x\sqrt{x}+C$ |
here |
| $\sqrt{x^2+1}$ |
$\displaystyle\frac{x}{2}\sqrt{x^2+1}+\frac{1}{2}\arsinh{x}+C$ |
here |
| $\sqrt{x^2-1}$ |
$\displaystyle\frac{x}{2}\sqrt{x^2-1}-\frac{1}{2}\arcosh{x}+C$ |
here |
| $\displaystyle\frac{1}{\sqrt{x^2+1}}$ |
$\arsinh{x}+C$ |
here |
| $\displaystyle\frac{1}{\sqrt{x^2-1}}$ |
$\arcosh{x}+C\;\; (x > 1)$ |
here |
Remark 2 The antiderivatives may be proven by differentiation; in some cases there are also given a link to a derivation.
Remark 3 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: $$ \int\limits_{-1}^1 |x|^{-1} \, dx=\frac{x\ln|x|}{|x|}\bigg|_{-1}^1=\frac{1\ln|1|}{|1|}-\frac{-1\ln|-1|}{|-1|}=0-0=0 $$
The above calculation is incorrect since $|x|^{-1}$ is not continuous at $x=0$ .
Instructions on how to add a function and its integral. Open the entry in edit mode. Using the appropriate table for your function (or make a new table if applicable), make a copy of the two lines of comment (starting with %) in the code (within the tabular environment) and paste it immediately before the comment. Uncomment the lines (take out the % symbols) after completing. Preview before saving the entry.
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"table of integrals" is owned by CWoo. [ full author list (4) ]
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Cross-references: integral, continuous at, domain, continuous, integrand, definite integral, link, differentiation, logarithmic integral, real number, Li, derivation, indefinite integrals, functions
This is version 39 of table of integrals, born on 2007-10-12, modified 2008-12-15.
Object id is 9991, canonical name is IntegralTables.
Accessed 2609 times total.
Classification:
| AMS MSC: | 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type) |
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Pending Errata and Addenda
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