Chapter 4 - Integrals - 4.5 The Substitution Rule - 4.5 Exercises - Page 346: 27

$\frac{sec^3(x)}{3} +C$

Evaluate the Integral using substitution: $\int sec^3(x)tan(x)dx$ Substitution Rule: $\int f(g(x))g’(x)dx = \int f(u)du$ $u= sec(x)$ $du =sec(x)tan(x)$ Solve the integral in terms of $u$: $\int u^2du$ $\frac{u^3}{3}+C$ Substitute for $u$: $\frac{sec^3(x)}{3} +C$