Integrals of Rational Functions
We found 28 formulas for integrals of rational functions
5.
$\int \left(ax+b\right)^n \,dx=\dfrac{\left(ax+b\right)^{n+1}}{a(n+1)}+C, n \ne 1$
8.
$\int \dfrac{ax+b}{cx+d}dx=\dfrac{a}{c}x+\dfrac{bc-ad}{c^2} \ln|cx+d|+C$
9.
$\int \dfrac{dx}{(x+a)(x+b)} dx=\dfrac{1}{a-b}+\ln\left| \dfrac{x+b}{x+a}\right|+C, a\ne b$
10.
$\int \dfrac{x}{a+bx}dx=\dfrac{1}{b^2}\left(a+bx-a\ln|a+bx|\right) + C$
11.
$\int \dfrac{x^2 dx}{a+bx}=\dfrac{1}{b^3}\left[ \dfrac{1}{2} (a+bx)^2 - 2a(a+bx)+a^2\ln|a+bx|\right]+C$
12.
$\int \dfrac{dx}{x(a+bx)}dx=\dfrac{1}{a} \ln \left| \dfrac{a+bx}{x} \right| + C$
13.
$\int \dfrac{dx}{x^2(a+bx)}dx=-\dfrac{1}{ax}+\dfrac{b}{a^2} \ln\left|\dfrac{a+bx}{x}\right|+C$
14.
$\int \dfrac{xdx}{\left(a+bx\right)^2}=\dfrac{1}{b^2} \left( \ln \left|a+bx\right|+\dfrac{a}{a+bx}\right) + C$
15.
$\int \dfrac{x^2}{\left(a+bx\right)^2}=\dfrac{1}{b^3} \left(a+bx-2a\ln|a+bx|-\dfrac{a^2}{a+bx}\right)+C$
16.
$\int \dfrac{dx}{x(a+bx)^2}=\dfrac{1}{a(a+bx)}+\dfrac{1}{a^2}\ln\left|\dfrac{a+bx}{x}\right|+C$
17.
$\int \dfrac{dx}{x^2-1}=\dfrac{1}{2}\ln\left|\dfrac{x-1}{x+1}\right|+C$
18.
$\int \dfrac{dx}{1-x^2}=\dfrac{1}{2}\ln\left|\dfrac{1+x}{1-x}\right|+C$
19.
$\int \dfrac{dx}{a^2-x^2}=\dfrac{1}{2a}\ln\left|\dfrac{a+x}{a-x}\right|+C$
20.
$\int \dfrac{dx}{x^2-a^2}=\dfrac{1}{2a}\ln\left|\dfrac{x-a}{x+a}\right|+C$
22.
$\int \dfrac{dx}{a^2+x^2}=\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+C$
23.
$\int \dfrac{dx}{a+bx^2}=\dfrac{1}{\sqrt{ab}}\arctan \left(x\sqrt{\dfrac{a}{b}}\right)+C, ab>0$
24.
$\int \dfrac{xdx}{a+bx^2}=\dfrac{1}{2b}\ln\left|x^2+\frac{a}{b}\right|+C$
25.
$\int \dfrac{dx}{x\left(a+bx^2\right)}=\dfrac{1}{2a}\ln\left|\dfrac{x^2}{a+bx^2}\right|+C$
26.
$\int \frac{dx}{a^2-b^2x^2}=\dfrac{1}{2ab}\ln\left|\dfrac{a+bx}{a-bx}\right|+C$
27.
$\int \dfrac{dx}{ax^2+bx+c}=\dfrac{1}{\sqrt{b^2-4ac}}\ln\left| \dfrac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}} \right|+C, b^2-4ac>0$
28.
$\int \dfrac{dx}{ax^2+bx+c}=\dfrac{2}{\sqrt{4ac-b^2}} \arctan \dfrac{2ax+b}{\sqrt{4ac-b^2}}+C$