integration formulas

Every student of calculus should aim to master each of the following integration forms. Treat \(u\) as a function of a real variable \(x\) and \(du\) as the derivative of \(u\) with respect to \(x\), i.e., as \(\frac{du}{dx}\). Where they appear, \(v\) is also a function of \(x\), \(c\) and \(a\) are constants, and \(e\) is the Euler number.

\[\begin{eqnarray*} \int cf(u)\,du & = & c\int f(u)\,du \\ & & \\ & & \\ \int (f(u)\pm g(u))\,du & = & \int f(u)\,du \pm \int g(u)\,du \\ & & \\ & & \\ \int \,du & = & u + c \\ & & \\ & & \\ \int u^n \,du & = & \frac{1}{n+1}u^{n+1} + c, n\neq -1 \\ & & \\ & & \\ \int \frac{1}{u}\,du & = & \ln|u| + c \\ & & \\ & & \\ \int e^u\,du & = & e^u + c \\ & & \\ & & \\ \int a^u\,du & = & \frac{a^u}{\ln a} + c \\ & & \\ & & \\ \int \sin u \,du & = & -\cos u + c \\ & & \\ & & \\ \int \cos u \,du & = & \sin u + c \\ & & \\ & & \\ \int \tan u \,du & = & -\ln|\cos u| + c \\ & & \\ & & \\ \int \cot u \,du & = & \ln|\sin u| + c \\ & & \\ & & \\ \int \sec u \,du & = & \ln|\sec u + \tan u| + c \\ & & \\ & & \\ \int \csc u \,du & = & -\ln|\csc u + \cot u| + c \\ & & \\ & & \\ \int \sec^2 u \,du & = & \tan u + c \\ & & \\ & & \\ \int \csc^2 u \,du & = & -\cot u + c \\ & & \\ & & \\ \int \sec u \tan u \,du & = & \sec u + c \\ & & \\ & & \\ \int \csc u \cot u \,du & = & -\csc u + c \\ & & \\ & & \\ \int \frac{\,du}{\sqrt{a^2-u^2}} & = & \arcsin \frac{u}{a} + c \\ & & \\ & & \\ \int \frac{\,du}{a^2+u^2} & = & \frac{1}{a} \arctan \frac{u}{a} + c \\ & & \\ & & \\ \int \frac{\,du}{u\sqrt{u^2-a^2}} & = & \frac{1}{a}\mbox{ arcsec } \frac{|u|}{a} + c \\ & & \\ & & \\ \int u\,dv & = & uv-\int v\,du \\ & & \\ & & \\ \end{eqnarray*} \]