Integrals of Rational Functions

Integrals involving \(ax + b\)

\( \int (ax+b)^n dx = \frac{(ax+b)^{n+1}}{a(n+1)}, \quad (\text{for } n \ne 1) \)

\( \int \frac{1}{ax+b}dx = \frac{1}{a}\ln|ax+b| \)

\( \int x (ax+b)^ndx = \frac{a(n+1)x-b}{a^2(n+1)(n+2)}(ax+b)^{n+1}, \quad (\text{for } n \ne -1, n\ne-2) \)

\( \int \frac{x}{ax+b}dx = \frac{x}{2} – \frac{b}{a^2}\ln|ax+b| \)

\( \int \frac{x}{(ax+b)^2}dx = \frac{b}{a^2(ax+b)} – \frac{1}{a^2}\ln|ax+b| \)

\( \int \frac{x^2}{ax+b} dx = \frac{1}{a^3} \left(\frac{(ax+b)^2}{2}-2b(ax+b)+b^2\ln|ax+b| \right) \)

\( \int \frac{x^2}{(ax+b)^2} dx = \frac{1}{a^3} \left(ax+b-2\,b\,\ln|ax+b| – \frac{b^2}{ax+b}\right) \)

\( \int \frac{x^2}{(ax+b)^3} dx = \frac{1}{a^3} \left( \ln|ax+b| + \frac{2b}{ax+b} – \frac{b^2}{2(ax+b)^2} \right) \)

\(
\int \frac{x^2}{(ax+b)^n} dx = \frac{1}{a^3} \left( -\frac{(ax+b)^{3-n}}{n-3} + \frac{2b(a+b)^{2-n}}{n-2} – \frac{b^2(ax+b)^{1-n}}{n-1}\right)
\)

\( \int \frac{1}{x(ax+b)}dx = -\frac{1}{b}\ln\left|\frac{ax+b}{x}\right| \)

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

\( \int \frac{1}{x^2(ax+b)^2}dx = -a\left(\frac{1}{b^2(ax+b)} + \frac{1}{ab^2x} – \frac{2}{b^3}\ln\left|\frac{ax+b}{x} \right|\right) \)

Integrals involving \(ax^2 + bx + c\)

\( \frac{1}{x^2+a^2} dx = \frac{1}{a}\arctan\frac{x}{a} \)

\( \frac{1}{x^2 – a^2} dx = \frac{1}{2a} \ln\left| \frac{x-a}{x+a}\right| \)

\(
\int \frac{1}{ax^2 + bx + c} dx = \left\{
\begin{aligned}
& \frac{2}{\sqrt{4ac – b^2}} \arctan \frac{2ax + b}{\sqrt{4ac-b^2}} \quad \text{for } 4ac – b^2 > 0 \\
& \frac{2}{\sqrt{b^2 – 4ac}} \ln \left| \frac{2ax + b – \sqrt{b^2-4ac}}{2ax + b + \sqrt{b^2-4ac}} \right| \quad \text{for } 4ac – b^2 < 0 \\
& -\frac{2}{2ax+b} \quad \text{for } 4ac – b^2 = 0
\end{aligned} \right.
\)

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

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

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

 

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