Chapter 7.2, Problem 72ES

In each part, use integration by parts or other methods to derive the reduction formula.

$\begin{array}{l}\text{(a)}{\displaystyle \int {\sec }^{n}xdx=\frac{{\sec }^{n-2}x\tan x}{n-1}}+\frac{n-2}{n-1}{\displaystyle \int {\sec }^{n-2}}xdx\\ \text{(b)}{\displaystyle \int {\tan }^{n}xdx=\frac{{\tan }^{n-1}x}{n-1}-{\displaystyle \int {\tan }^{n-2}xdx}}\\ \text{(c)}{\displaystyle \int x^n{e}^{x}dx=x^n{e}^x-n{{\displaystyle \int x}}^{n-1}{e}^{x}dx}\end{array}$