Useful derivatives (or integrals, depends on how you look at it)

$$\begin{align*} \frac{\mathrm{d} }{\mathrm{d} x}\ln |\sec x| &= \tan x \\ \frac{\mathrm{d} }{\mathrm{d} x}\ln |\sin x| &= \cot x \\ \frac{\mathrm{d} }{\mathrm{d} x} \ln |\sec x + \tan x| &= \sec x \\ \frac{\mathrm{d} }{\mathrm{d} x} \frac{a^x}{\ln a} &= a^x \\ \frac{\mathrm{d} }{\mathrm{d} x} (\frac{1}{a} \tan^{-1} \frac{x}{a}) &= \frac{1}{a^2 + x^2} \\ \frac{\mathrm{d} }{\mathrm{d} x} \sin^{-1}\frac{x}{a} &= \frac{1}{\sqrt{a^2 - x^2}} \\ \frac{\mathrm{d} }{\mathrm{d} x} \frac{1}{a} \ln |ax + b| &= \frac{1}{ax + b} \\ \frac{\mathrm{d} }{\mathrm{d} x} \sqrt{ax^2 + b} &= \frac{ax}{\sqrt{ax^2 + b}} \\ \frac{\mathrm{d} }{\mathrm{d} x} \frac{1}{2a} \ln \left|\frac{x-a}{x+a}\right| &= \frac{1}{x^2 - a^2} \end{align*}$$