Indefinite Integral

Tags

Calculus

Cegep/2

Word count

344 words

Reading time

3 minutes

General form of all antiderivatives of a function

Properties

• $\int 0dx=C$
• $\int kf(x)dx=k\int f(x)dx$
• $\int (f(x)\pm g(x))dx=\int f(x)dx\pm \int g(x)dx$
• $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for $n\ne -1$
• $\int x^{-1}dx=\ln |x|+c$ for $x\ne 0$
• $\int a^{x}dx=\frac{a^x}{\ln a}+C$
• $\int \cos x\mathrm{d}x=\sin x$
• $\int \sin x\mathrm{d}x=-\cos x$
• $\int {\sec }^{2}x\mathrm{d}x=\tan x$
• $\int \tan x\sec x\mathrm{d}x=\sec x$
• $\int \cot x\csc x\mathrm{d}x=-\csc x$
• $\int {\csc }^{2}x\mathrm{d}x=-\cot x$
• $\int \frac{1}{\sqrt{1-x^2}}\mathrm{d}x=\arcsin x=-\arccos x$
• $\int \frac{1}{|x|\sqrt{x^2-1}}\mathrm{d}x=\mathrm{arcsec}x=-\mathrm{arccsc}x$
• $\int \frac{1}{1+x^2}\mathrm{d}x=\arctan x=-\mathrm{arccot}x$
• ==$\int \ln xdx=x\ln x-x+C$==
• ==$\int \tan xdx=\ln \sec |x|+C$==
• ==$\int \sec xdx=\ln |\sec x+\tan x|+C$==

Examples

Evaluate the following indefinite integrals.

$$\int (\cos x+5\sin x-3^x)dx=\sin x-5\cos x-\frac{3^x}{\ln 3}+C$$$$\int \frac{x^2-3\sqrt{x}+2x^{\frac{3}{2}}}{2x^{\frac{3}{2}}}dx=\frac{1}{2}\int (x^{\frac{1}{2}}-3x^{-1}+2)dx=\frac{1}{2}(\frac{x^{\frac{3}{2}}}{\frac{3}{2}}-3ln|x|+2x)+C$$$$\int 2x\cos (x^2+3)dx=\sin (x^2+3)+C$$

$\int {\sin }^{4}x\mathrm{d}x$

$$\begin{array}{rl}\int {\sin }^{4}x\mathrm{d}x& =\int {(\frac{1}{2}(1-\cos (2x)))}^2\mathrm{d}x\\ & =\frac{1}{4}\int (1-2\cos (2x)+{\cos }^2(2x))\mathrm{d}x\\ & =\frac{1}{4}\int (1-2\cos (2x)+\frac{1}{2}(1+\cos (2x)))\mathrm{d}x\\ & =\frac{1}{4}\int (\frac{3}{2}-\frac{3}{4}\cos (2x))\mathrm{d}x\\ & =\frac{1}{4}(\frac{3}{2}x-\frac{3}{8}\sin (2x))+C\\ & =\frac{3}{8}x-\frac{3}{32}\sin (2x)+C\end{array}$$

Contributors

Ajitani Hifumi

Changelog

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