❦ Integration

Function	Integral
\( x^n, n\neq -1 \)	\( \frac{x^{n+1}}{n+1}+C \)
\( n^x \)	\( \frac{n^x}{\ln n} + C \)
\( \frac{1}{x} \)	\( \ln \|x\| + \begin{cases} C_1 & \text{if } x\gt 0 \\ C_2 & \text{if } x\lt 0 \end{cases} \)
\( \sec ^2 x \)	\( \tan x + C\)
\( \frac{1}{\sqrt{x^2 + a^2}} \)	\( +C\)
\( \)	\( +C\)
\( \ln x \)	\( x \ln x - x +C\)
\( \)	\( +C\)

Function	Integral
\( \int_0^\infty x^{2n} e^{\frac{-x^2}{a^2}} \text d x \)	\( \sqrt \pi \frac{(2n)!}{n!} (\frac{a}{2})^{2n+1} \)
\( \int_0^\infty x^{2n+1} e^{\frac{-x^2}{a^2}} \text d x \)	\( \frac{n!}{2} a^{2n+2} \)
\( \int_{-\infty}^{\infty} e^{-x^2} \text d x \)	\( \sqrt \pi \)
\( \int_{0}^{\infty} \sqrt{x} e^{-x} \text d x \)	\( \frac{\sqrt \pi}{2} \)
\( \int \ln (1 + e^{-x}) \text d x \)	\( \frac{\pi^2}{12} \)
\( \int_0^\infty \frac{\sin x}{x} \text d x \)	\( \frac{\pi}{2} \)