Table of Indefinite Integrals

This is a table of common antiderivatives and definite integrals for the Calculus students. Remember after each indefinite integral there is a general constant of integration (+C).

Antiderivatives

Elementary

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\int x^n \, dx = \frac{1}{n+1} x^{n+1}, \, n \neq -1 </math>

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\int \frac{1}{x} \, dx = \ln|x| </math>

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\int e^x \, dx = e^x </math>

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\int \sin x \, dx = -\cos x </math>

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\int \cos x \, dx = \sin x </math>

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\int \tan x \, dx = - \ln \cos x </math>

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\int \sinh x \, dx = \cosh x </math>

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\int \cosh x \, dx = \sinh x </math>

Commonly encountered

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\int \ln x \, dx = x \ln x - x </math>

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\int a^x \, dx = \frac{1}{\ln a} a^x </math>

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\int \frac{1}{x^2+a^2} \, dx = \frac{1}{a} \tan^{-1} \frac{x}{a} = \frac{1}{a} \arctan \frac{x}{a} </math>

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\int \frac{1}{a^2-x^2} \, dx = \frac{1}{2} \ln \left| \frac{x+a}{x-a} \right| </math>

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\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \sin^{-1} \frac{x}{a} = \arcsin \frac{x}{a} </math>

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\int \frac{1}{\sqrt{x^2-a^2}} \, dx = \cosh^{-1} \frac{x}{a} = \ln(x+\sqrt{x^2-a^2}) </math>

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\int \frac{1}{\sqrt{x^2+a^2}} \, dx = \sinh^{-1} \frac{x}{a} = \ln(x+\sqrt{x^2+a^2}) </math>

See also

• Integration
• Methods of integration

External links

• Integral table