# Table of Integrals

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1. $$\int \frac{1}{x(a x+b)} d x=\frac{1}{b} \ln \left|\frac{x}{a x+b}\right|+C$$
2. $$\int \frac{x}{a x+b} d x=\frac{x}{a}-\frac{b}{a^{2}} \ln |a x+b|+C$$
3. $$\int \frac{1}{(x+a)(x+b)} d x=\frac{1}{b-a}(\ln |x+a|-\ln |x+b|)+C=\frac{1}{b-a} \ln \left|\frac{x+a}{x+b}\right|+C, \quad a \neq b$$
4. $$\int \frac{1}{a^{2}-x^{2}} d x=\frac{1}{2 a} \ln \left|\frac{x+a}{x-a}\right|+C$$
5. $$\int \frac{1}{\sqrt{x^{2} \pm a^{2}}} d x=\ln \left|x+\sqrt{x^{2} \pm a^{2}}\right|+C$$
6. $$\int \sqrt{x^{2} \pm a^{2}} d x=\frac{x}{2} \sqrt{x^{2} \pm a^{2}}+\frac{a^{2}}{2} \ln \left|x+\sqrt{x^{2} \pm a^{2}}\right|+C$$
7. $$\int x(a x+b)^{n} d x=\frac{(a x+b)^{n+1}}{a}\left(\frac{a x+b}{n+2}-\frac{b}{n+1}\right)+C, \quad n \neq-1,-2$$
8. $$\int x^{n} e^{a x} d x=\frac{1}{a} x^{n} e^{a x}-\frac{n}{a} \int x^{n-1} e^{a x} d x+C$$