Answer
$\frac{\sec^7x}{7} - \frac{\sec^5x}{5}+C$
Work Step by Step
Find the indefinite integral
$\int sec^5xtan^3xdx$
$\int sec^4xtan^2x secxtanxdx$
$\int sec^4x(sec^2x -1) secx tanx dx$
$ \int sec^6x secxtanxdx - \int sec^4xsecxtanxdx$
Use u-substitution, let $u=secx$, $du= secxtanxdx$
$\int u^6du - \int u^4du$, Integrate
$ \frac{u^7}{7} - \frac{u^5}{5} +C$
$\frac{\sec^7x}{7} - \frac{\sec^5x}{5}+C$
