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Table of Indefinite Integrals (view source)
Revision as of 18:00, December 19, 2018
, 18:00, December 19, 2018→Antiderivatives: Indent math expression
==Antiderivatives==
==Antiderivatives==
===Elementary===
===Elementary===
<math>
:<math>
\int x^n \, dx = \frac{1}{n+1} x^{n+1}, \, n \neq -1
\int x^n \, dx = \frac{1}{n+1} x^{n+1}, \, n \neq -1
</math>
</math>
<math>
:<math>
\int \frac{1}{x} \, dx = \ln|x|
\int \frac{1}{x} \, dx = \ln|x|
</math>
</math>
<math>
:<math>
\int e^x \, dx = e^x
\int e^x \, dx = e^x
</math>
</math>
<math>
:<math>
\int \sin x \, dx = -\cos x
\int \sin x \, dx = -\cos x
</math>
</math>
<math>
:<math>
\int \cos x \, dx = \sin x
\int \cos x \, dx = \sin x
</math>
</math>
<math>
:<math>
\int \tan x \, dx = - \ln \cos x
\int \tan x \, dx = - \ln \cos x
</math>
</math>
<math>
:<math>
\int \sinh x \, dx = \cosh x
\int \sinh x \, dx = \cosh x
</math>
</math>
<math>
:<math>
\int \cosh x \, dx = \sinh x
\int \cosh x \, dx = \sinh x
</math>
</math>
===Commonly encountered===
===Commonly encountered===
<math>
:<math>
\int \ln x \, dx = x \ln x - x
\int \ln x \, dx = x \ln x - x
</math>
</math>
<math>
:<math>
\int a^x \, dx = \frac{1}{\ln a} a^x
\int a^x \, dx = \frac{1}{\ln a} a^x
</math>
</math>
<math>
:<math>
\int \frac{1}{x^2+a^2} \, dx = \frac{1}{a} \tan^{-1} \frac{x}{a} = \frac{1}{a} \arctan \frac{x}{a}
\int \frac{1}{x^2+a^2} \, dx = \frac{1}{a} \tan^{-1} \frac{x}{a} = \frac{1}{a} \arctan \frac{x}{a}
</math>
</math>
<math>
:<math>
\int \frac{1}{a^2-x^2} \, dx = \frac{1}{2} \ln \left| \frac{x+a}{x-a} \right|
\int \frac{1}{a^2-x^2} \, dx = \frac{1}{2} \ln \left| \frac{x+a}{x-a} \right|
</math>
</math>
<math>
:<math>
\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \sin^{-1} \frac{x}{a} = \arcsin \frac{x}{a}
\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \sin^{-1} \frac{x}{a} = \arcsin \frac{x}{a}
</math>
</math>
<math>
:<math>
\int \frac{1}{\sqrt{x^2-a^2}} \, dx = \cosh^{-1} \frac{x}{a} = \ln(x+\sqrt{x^2-a^2})
\int \frac{1}{\sqrt{x^2-a^2}} \, dx = \cosh^{-1} \frac{x}{a} = \ln(x+\sqrt{x^2-a^2})
</math>
</math>
<math>
:<math>
\int \frac{1}{\sqrt{x^2+a^2}} \, dx = \sinh^{-1} \frac{x}{a} = \ln(x+\sqrt{x^2+a^2})
\int \frac{1}{\sqrt{x^2+a^2}} \, dx = \sinh^{-1} \frac{x}{a} = \ln(x+\sqrt{x^2+a^2})
</math>
</math>