Table of integrals

The table below contains integration results used throughout this notes.

Indefinite integrals

$$\begin{array}{rll}\int x^{n}dx& =\frac{1}{n+1}{x}^{n+1}& \\ \int udv& =uv-\int vdu& \text{Integration by parts}\\ \int e^{\alpha x}dx& =\frac{1}{a}{e}^{\alpha x}& \\ \int \ln (x)dx& =x\left(\ln (x)-1\right)& \\ \int si{n}^2(\alpha x)& =\frac{x}{2}-\frac{1}{4a}\sin (2ax)& \end{array}$$

where

• $\ln (x):={\log }_e(x)$ is the natural logarithm.

Integrals over $\mathbb{R}$

$$\begin{array}{rl}{\int }_{\mathbb{R}}{e}^{-\alpha x^2}& =\sqrt{\frac{\pi }{\alpha }}\end{array}$$

Integrals over ${\mathbb{R}}^{+}$

$$\begin{array}{rl}{\int }_{{\mathbb{R}}^{+}}{e}^{-\alpha x^2}& =\frac{1}{2}\sqrt{\frac{\pi }{\alpha }}\\ {\int }_{{\mathbb{R}}^{+}}{x}^n{e}^{-\alpha x^2}& =(...)\end{array}$$