Integral with $ax^2+b$

22

$$\int \frac{1}{ax^2+b} \ \mathrm{d}x = \left\{\begin{aligned} &\frac{1}{\sqrt{ab}}\arctan\sqrt{\frac{a}{b}}x+C \quad (b>0)\\ &\frac{1}{2\sqrt{-ab}}\ln\left|\frac{\sqrt{a}x-\sqrt{-b}}{\sqrt{a}x+\sqrt{-b}}\right|+C \quad (b<0) \end{aligned} \right.$$

Derivation

23

$$\int\frac{x}{ax^2+b}\ \mathrm{d}x=\frac{1}{2a}\ln\left|ax^2+b\right|+C$$

Derivation

24

$$\int \frac{x^2}{ax^2+b}\ \mathrm{d}x=\frac{x}{a}-\frac{b}{a}\int \frac{1}{ax^2+b}\ \mathrm{d}x$$

Derivation

25

$$\int \frac{1}{x\left(ax^2+b\right)}\ \mathrm{d}x=\frac{1}{2b}\ln \frac{x^2}{\left| ax^2+b\right|}+C$$

Derivation

26

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

Derivation

27

$$\int \frac{1}{x^3\left(ax^2+b\right)}\ \mathrm{d}x=\frac{a}{2b^2}\ln{\frac{\left|ax^2+b\right|}{x^2}}-\frac{1}{2bx^2}+C$$

Derivation

28

$$\int \frac{1}{\left(ax^2+b\right)^2}\ \mathrm{d}x=\frac{x}{2b\left(ax^2+b\right)}+\frac{1}{2b}\int \frac{1}{ax^2+b}\ \mathrm{d}x$$

Derivation