Section C.1 Integrals Involving Exponential or Trigonometric Functions
\begin{align}
\int u^n e^{au}\ du \amp=\amp \frac{1}{a} u^n e^{au} - \frac{n}{a}\int u^{n-1} e^{au}\ du\tag{C.1.1}\\
\int \sin^n u\ du \amp=\amp -\frac{1}{n}\sin^{n-1}u \cos u + \frac{n-1}{n}\int \sin^{n-2} u\ du\tag{C.1.2}\\
\int \cos^n u\ du \amp=\amp \frac{1}{n}\cos^{n-1}u \sin u + \frac{n-1}{n}\int \cos^{n-2} u\ du\tag{C.1.3}\\
\int \tan^n u\ du \amp=\amp \frac{1}{n-1}\tan^{n-1}u - \int \tan^{n-2} u\ du\tag{C.1.4}\\
\int \cot^n u\ du \amp=\amp \frac{-1}{n-1}\cot^{n-1}u - \int \cot^{n-2} u\ du\tag{C.1.5}\\
\int \sec^n u\ du \amp=\amp \frac{1}{n-1}\tan u \sec^{n-2}u + \frac{n-2}{n-1}\int \sec^{n-2} u\ du\tag{C.1.6}\\
\int \csc^n u\ du \amp=\amp \frac{-1}{n-1}\cot u \csc^{n-2}u + \frac{n-2}{n-1}\int \csc^{n-2} u\ du\tag{C.1.7}\\
\int u^n \sin u\ du \amp=\amp -u^n \cos u + n \int u^{n-1} \cos u\ du\tag{C.1.8}\\
\int u^n \cos u\ du \amp=\amp u^n \sin u - n \int u^{n-1} \sin u\ du\tag{C.1.9}
\end{align}
\begin{align}
\int \sin^n u \cos^m u\ du \amp=\amp \int (1 - \cos^2 u)^k \cos^n u \sin u \ du = -\int (1 - v^2)^k v^n\ dv,\, m = 2k+1,\, v = \cos u\tag{C.1.10}\\
\int \sin^n u \cos^m u\ du \amp=\amp \int (1 - \sin^2 u)^k \sin^m u \cos u\ du = \int (1 - v^2)^k v^m\ dv,\, n = 2k+1,\, v = \sin u\tag{C.1.11}\\
\int \sin^n u \cos^m u\ du \amp=\amp
\int \left( \frac{1 - \cos 2u}{2} \right)^{k_1} \left( \frac{1 + \cos 2u}{2} \right)^{k_2},\, n = 2 k_1,\, m = 2 k_2\tag{C.1.12}
\end{align}
\begin{align}
\int \sin^n u \cos^m u\ du \amp=\amp -\frac{\sin^{n-1}u \cos^{m+1}u}{n+m} - \frac{n-1}{n+m} \int \sin^{n-2}u \cos^{m}u\ du,\, n \geq 2\tag{C.1.13}\\
\int \sin^n u \cos^m u\ du \amp=\amp \frac{\sin^{n+1}u \cos^{m-1}u}{n+m} + \frac{m-1}{n+m} \int \sin^{n}u \cos^{m-2}u\ du,,\, m \geq 2\tag{C.1.14}
\end{align}
