List of integrals of rational functions

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The following is a list of integrals (antiderivative functions) of rational functions. For a more complete list of integrals, please see table of integrals and list of integrals.

$\int (ax + b)^n dx$	$= \frac{(ax + b)^{n+1}}{a(n + 1)} \qquad\mbox{(for } n\neq -1\mbox{)}\,\!$
$\int\frac{dx}{ax + b}$	$= \frac{1}{a}\ln\left\|ax + b\right\|$
$\int x(ax + b)^n dx$	$= \frac{a(n + 1)x - b}{a^2(n + 1)(n + 2)} (ax + b)^{n+1} \qquad\mbox{(for }n \not\in \{-1, -2\}\mbox{)}$

$\int\frac{x dx}{ax + b}$	$= \frac{x}{a} - \frac{b}{a^2}\ln\left\|ax + b\right\|$
$\int\frac{x dx}{(ax + b)^2}$	$= \frac{b}{a^2(ax + b)} + \frac{1}{a^2}\ln\left\|ax + b\right\|$
$\int\frac{x dx}{(ax + b)^n}$	$= \frac{a(1 - n)x - b}{a^2(n - 1)(n - 2)(ax + b)^{n-1}} \qquad\mbox{(for } n\not\in \{1, 2\}\mbox{)}$

$\int\frac{x^2 dx}{ax + b}$	$= \frac{1}{a^3}\left(\frac{(ax + b)^2}{2} - 2b(ax + b) + b^2\ln\left\|ax + b\right\|\right)$
$\int\frac{x^2 dx}{(ax + b)^2}$	$= \frac{1}{a^3}\left(ax + b - 2b\ln\left\|ax + b\right\| - \frac{b^2}{ax + b}\right)$
$\int\frac{x^2 dx}{(ax + b)^3}$	$= \frac{1}{a^3}\left(\ln\left\|ax + b\right\| + \frac{2b}{ax + b} - \frac{b^2}{2(ax + b)^2}\right)$
$\int\frac{x^2 dx}{(ax + b)^n}$	$= \frac{1}{a^3}\left(-\frac{(ax + b)^{3-n}}{(n-3)} + \frac{2b (a + b)^{2-n}}{(n-2)} - \frac{b^2 (ax + b)^{1-n}}{(n - 1)}\right) \qquad\mbox{(for } n\not\in \{1, 2, 3\}\mbox{)}$

$\int\frac{dx}{x(ax + b)}$	$= -\frac{1}{b}\ln\left\|\frac{ax+b}{x}\right\|$
$\int\frac{dx}{x^2(ax+b)}$	$= -\frac{1}{bx} + \frac{a}{b^2}\ln\left\|\frac{ax+b}{x}\right\|$
$\int\frac{dx}{x^2(ax+b)^2}$	$= -a\left(\frac{1}{b^2(ax+b)} + \frac{1}{ab^2x} - \frac{2}{b^3}\ln\left\|\frac{ax+b}{x}\right\|\right)$
$\int\frac{dx}{x^2+a^2}$	$= \frac{1}{a}\arctan\frac{x}{a}\,\!$
$\int\frac{dx}{x^2-a^2} =$	$-\frac{1}{a}\,\mathrm{arctanh}\frac{x}{a} = \frac{1}{2a}\ln\frac{a-x}{a+x} \qquad\mbox{(for }\|x\| < \|a\|\mbox{)}\,\!$
	$-\frac{1}{a}\,\mathrm{arccoth}\frac{x}{a} = \frac{1}{2a}\ln\frac{x-a}{x+a} \qquad\mbox{(for }\|x\| > \|a\|\mbox{)}\,\!$

$\int\frac{dx}{ax^2+bx+c} =$	$\frac{2}{\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(for }4ac-b^2>0\mbox{)}$
	$-\frac{2}{\sqrt{b^2-4ac}}\,\mathrm{arctanh}\frac{2ax+b}{\sqrt{b^2-4ac}} = \frac{1}{\sqrt{b^2-4ac}}\ln\left\|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right\| \qquad\mbox{(for }4ac-b^2<0\mbox{)}$
	$-\frac{2}{2ax+b}\qquad\mbox{(for }4ac-b^2=0\mbox{)}$
$\int\frac{x dx}{ax^2+bx+c}$	$= \frac{1}{2a}\ln\left\|ax^2+bx+c\right\|-\frac{b}{2a}\int\frac{dx}{ax^2+bx+c}$

$\int\frac{(mx+n) dx}{ax^2+bx+c} =$	$\frac{m}{2a}\ln\left\|ax^2+bx+c\right\|+\frac{2an-bm}{a\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(for }4ac-b^2>0\mbox{)}$
	$\frac{m}{2a}\ln\left\|ax^2+bx+c\right\|+\frac{2an-bm}{a\sqrt{b^2-4ac}}\,\mathrm{artanh}\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad\mbox{(for }4ac-b^2<0\mbox{)}$
	$\frac{m}{2a}\ln\left\|ax^2+bx+c\right\|-\frac{2an-bm}{a(2ax+b)} \,\,\,\,\,\,\,\,\,\, \qquad\mbox{(for }4ac-b^2=0\mbox{)}$

$\int\frac{dx}{(ax^2+bx+c)^n} = \frac{2ax+b}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}+\frac{(2n-3)2a}{(n-1)(4ac-b^2)}\int\frac{dx}{(ax^2+bx+c)^{n-1}}\,\!$

$\int\frac{x dx}{(ax^2+bx+c)^n} = \frac{bx+2c}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}-\frac{b(2n-3)}{(n-1)(4ac-b^2)}\int\frac{dx}{(ax^2+bx+c)^{n-1}}\,\!$

$\int\frac{dx}{x(ax^2+bx+c)} = \frac{1}{2c}\ln\left|\frac{x^2}{ax^2+bx+c}\right|-\frac{b}{2c}\int\frac{dx}{ax^2+bx+c}$

Any rational function can be integrated using above equations and partial fractions in integration, by decomposing the rational function into a sum of functions of the form:

$\frac{ex + f}{\left(ax^2+bx+c\right)^n}$ .

Lists of integrals
Rational functions • Irrational functions • Trigonometric functions • Inverse trigonometric functions • Hyperbolic functions • Arc hyperbolic functions • Exponential functions • Logarithmic functions • Arc functions

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Categories: Integrals | Mathematics-related lists

$\int\frac{(mx+n) dx}{ax^2+bx+c} =$	$\frac{m}{2a}\ln\left\|ax^2+bx+c\right\|+\frac{2an-bm}{a\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad\mbox{(for }4ac-b^2>0\mbox{)}$
	$\frac{m}{2a}\ln\left\|ax^2+bx+c\right\|+\frac{2an-bm}{a\sqrt{b^2-4ac}}\,\mathrm{artanh}\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad\mbox{(for }4ac-b^2<0\mbox{)}$
	$\frac{m}{2a}\ln\left\|ax^2+bx+c\right\|-\frac{2an-bm}{a(2ax+b)} \,\,\,\,\,\,\,\,\,\, \qquad\mbox{(for }4ac-b^2=0\mbox{)}$

List of integrals of rational functions

From Wikipedia, the free encyclopedia

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