A.4: Table of Integrals

CLP-3 Multivariable Calculus · LibreTexts

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    Throughout this table, \(a\) and \(b\) are given constants, independent of \(x\) and \(C\) is an arbitrary constant.

    \(f(x)\) \(F(x)=\int f(x)\ \mathrm{d}{x} \)
    \(af(x)+bg(x)\) \(a\int f(x)\ \mathrm{d}{x} +b\int g(x)\ \mathrm{d}{x} \ +\ C\)
    \(f(x)+g(x)\) \(\int f(x)\ \mathrm{d}{x} +\int g(x)\ \mathrm{d}{x} \ +\ C\)
    \(f(x)-g(x)\) \(\int f(x)\ \mathrm{d}{x} -\int g(x)\ \mathrm{d}{x} \ +\ C\)
    \(af(x)\) \(a\int f(x)\ \mathrm{d}{x} \ +\ C\)
    \(u(x)v'(x)\) \(u(x)v(x)-\int u'(x)v(x)\ \mathrm{d}{x} \ +\ C\)
    \(f\big(y(x)\big)y'(x)\) \(F\big(y(x)\big)\hbox{ where }F(y)=\int f(y)\ \mathrm{d}{y} \)
    \(a\) \(ax+C\)
    \(x^a\) \(\frac{x^{a+1}}{a+1}+C\hbox{ if }a\ne-1\)
    \(\frac{1}{x}\) \(\ln|x|+C\)
    \(g(x)^ag'(x)\) \(\frac{g(x)^{a+1}}{a+1}+C\hbox{ if }a\ne -1\)

     

    \(f(x)\) \(F(x)=\int f(x)\ \mathrm{d}{x} \)
    \(\sin x\) \(-\cos x+C\)
    \(g'(x)\sin g(x)\) \(-\cos g(x)+C\)
    \(\cos x\) \(\sin x+C\)
    \(\tan x\) \(\ln|\sec x|+C\)
    \(\csc x\) \(\ln |\csc x-\cot x|+C\)
    \(\sec x\) \(\ln |\sec x+\tan x|+C\)
    \(\cot x\) \(\ln|\sin x|+C\)
    \(\sec^2 x\) \(\tan x+C\)
    \(\csc^2 x\) \(-\cot x+C\)
    \(\sec x\tan x\) \(\sec x+C\)
    \(\csc x\cot x\) \(-\csc x+C\)

     

    \(f(x)\) \(F(x)=\int f(x)\ \mathrm{d}{x} \)
    \(e^x\) \(e^x+C\)
    \(e^{g(x)}g'(x)\) \(e^{g(x)}+C\)
    \(e^{ax}\) \(\frac{1}{a}\ e^{ax}+C\)
    \(a^x\) \(\frac{1}{\ln a}\ a^x+C\)
    \(\ln x\) \(x\ln x -x+C\)
    \(\frac{1}{\sqrt{1-x^2}}\) \(\arcsin x+C\)
    \(\frac{g'(x)}{\sqrt{1-g(x)^2}}\) \(\arcsin g(x)+C\)
    \(\frac{1}{\sqrt{a^2-x^2}}\) \(\arcsin \frac{x}{a}+C\)
    \(\frac{1}{1+x^2}\) \(\arctan x+C\)
    \(\frac{g'(x)}{1+g(x)^2}\) \(\arctan g(x)+C\)
    \(\frac{1}{a^2+x^2}\) \(\frac{1}{a}\arctan \frac{x}{a}+C\)
    \(\frac{1}{x\sqrt{x^2-1}}\) \(\textrm{arcsec} x+C\) \quad(\(x \gt 1\))

    This page titled A.4: Table of Integrals was last modified on Sat, 26 Feb 2022 13:59:04 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform.