Common Integrals INTEGRALS CONTAINING ax+b

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

8.

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

9.

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

10.

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

11.

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

12.

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

13.

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

14.

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

15.

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

16.

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

17.

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

18.

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

19.

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

20.

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

21.

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

22.

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

23.

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

24.

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

25.

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

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