Common Integrals INTEGRALS CONTAINING ax+b

1.
$\displaystyle \int\displaystyle \frac{dx}{ax+b}=\displaystyle \frac{1}{a}\ln(ax+b)$

2.
$\displaystyle \int\displaystyle \frac{x\,dx}{ax+b}=\displaystyle \frac{x}{a}-\displaystyle \frac{b}{a^{2}}\ln(ax+b)$

3.
$\displaystyle \int\displaystyle \frac{x^3\,dx}{ax+b}=\displaystyle \frac{(ax+b)...
...splaystyle \frac{3b^{2}(ax+b)}{a^{4}}-
\displaystyle \frac{b^{3}}{a^4}\ln(ax+b)$

4.
$\displaystyle \int\displaystyle \frac{x^2\,dx}{ax+b}=\displaystyle \frac{(ax+b)...
...\displaystyle \frac{2b(ax+b)}{a^{3}}+\displaystyle \frac{b^{2}}{a^{3}}\ln(ax+b)$

5.
$\displaystyle \int\displaystyle \frac{dx}{x(ax+b)}=\displaystyle \frac{1}{b}\ln\left(\displaystyle \frac{x}{ax+b}\right)$

6.
$\displaystyle \int\displaystyle \frac{dx}{x^{2}(ax+b)}=-\displaystyle \frac{1}{bx}+\displaystyle \frac{a}{b^{2}}\ln\left(\displaystyle \frac{ax+b}{x}\right)$

7.
$\displaystyle \int\displaystyle \frac{dx}{x^{3}(ax+b)}=\displaystyle \frac{2ax-...
...}+\displaystyle \frac{a^{2}}{b^{3}}\ln\left(\displaystyle \frac{x}{ax+b}\right)$

8.
$\displaystyle \int\displaystyle \frac{dx}{(ax+b)^{2}}=\displaystyle \frac{-1}{a(ax+b)}$

9.
$\displaystyle \int\displaystyle \frac{x\,dx}{(ax+b)^{2}}=\displaystyle \frac{b}{a^{2}(ax+b)}+\displaystyle \frac{1}{a^{2}}\ln(ax+b)$

10.
$\displaystyle \int\displaystyle \frac{x^{2}\,dx}{(ax+b)^{2}}=\displaystyle \fra...
...\displaystyle \frac{b^{2}}{a^{3}(ax+b)}-\displaystyle \frac{2b}{a^{3}}\ln(ax+b)$

11.
$\displaystyle \int\displaystyle \frac{x^{3}\,dx}{(ax+b)^{2}}=\displaystyle \fra...
...playstyle \frac{b^{3}}{a^{4}(ax+b)}+\displaystyle \frac{3b^{2}}{a^{4}}\ln(ax+b)$

12.
$\displaystyle \int\displaystyle \frac{dx}{x(ax+b)}=\displaystyle \frac{1}{b(ax+b)}+\displaystyle \frac{1}{b^{2}}\ln\left(\displaystyle \frac{x}{ax+b}\right)$

13.
$\displaystyle \int\displaystyle \frac{dx}{x^{2}(ax+b)^{2}}=\displaystyle \frac{...
...2}x}+\displaystyle \frac{2a}{b^{3}}\ln\left(\displaystyle \frac{ax+b}{x}\right)$

14.
$\displaystyle \int\displaystyle \frac{dx}{x^{3}(ax+b)^{2}}=-\displaystyle \frac...
...-\displaystyle \frac{3a^{2}}{b^{4}}\ln\left(\displaystyle \frac{ax+b}{x}\right)$

15.
$\displaystyle \int\displaystyle \frac{dx}{(ax+b)^{3}}=\displaystyle \frac{-1}{2(ax+b)^{2}}$

16.
$\displaystyle \int\displaystyle \frac{x\,dx}{(ax+b)^{3}}=\displaystyle \frac{-1}{a^{2}(ax+b)}+\displaystyle \frac{b}{2a^{2}(ax+b)^{2}}$

17.
$\displaystyle \int\displaystyle \frac{x^{2}\,dx}{(ax+b)^{3}}=\displaystyle \fra...
...playstyle \frac{b^{2}}{2a^{3}(ax+b)^{2}}+\displaystyle \frac{1}{a^{3}}\ln(ax+b)$

18.
$\displaystyle \int\displaystyle \frac{x^{3}\,dx}{(ax+b)^{3}}=\displaystyle \fra...
...splaystyle \frac{b^3}{2a^{4}(ax+b)^{2}}-\displaystyle \frac{3b}{a^{4}}\ln(ax+b)$

19.
$\displaystyle \int\displaystyle \frac{dx}{x(ax+b)^{3}}=\displaystyle \frac{a^{2...
...x+b)}-\displaystyle \frac{1}{b^{3}}\ln\left(\displaystyle \frac{ax+b}{x}\right)$

20.
$\displaystyle \int\displaystyle \frac{dx}{x^{2}(ax+b)^{3}}=\displaystyle \frac{...
...3}x}+\displaystyle \frac{3a}{b^{4}}\ln\left(\displaystyle \frac{ax+b}{x}\right)$

21.
$\displaystyle \int\displaystyle \frac{dx}{x^{3}(ax+b)^{3}}=\displaystyle \frac{...
...-\displaystyle \frac{6a^{2}}{b^{5}}\ln\left(\displaystyle \frac{ax+b}{x}\right)$

22.
$\displaystyle \int(ax+b)^{\displaystyle n}\,dx=\displaystyle \frac{(ax+b)^{\displaystyle n+1}}{(n+1)a}\, ,\;\; n\neq -1$

23.
$\displaystyle \int x(ax+b)^{\displaystyle n}\,dx = \displaystyle \frac{(ax+b)^{...
...e \frac{b(ax+b)^{\displaystyle n+1}}{(n+1)a^{\displaystyle2}},\;\;\;n\neq -1,-2$

24.
$\displaystyle \int x^{\displaystyle2}(ax+b)^{\displaystyle n}\,dx=\displaystyle...
...ystyle2}(ax+b)^{\displaystyle n+1}}{(n+1)a^{\displaystyle3}}\;,\;n\neq -1,-2,-3$

25.

$\displaystyle \int x^{\displaystyle m}(ax+b)^{\displaystyle n}\,dx=\left\{ \beg...
... \int x^{\displaystyle m}(ax+b)^{\displaystyle{n+1}}\,dx
\end{array} \right. $

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