Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 30

$$\frac{1}{5}\sec^5y+C$$

Given $$\int \sec ^{5} y \tan y d y$$ Let $$ \sec y \to \sec y \tan y d y$$ Then \begin{align*} \int \sec ^{5} y \tan y d y&=\int \sec ^{4} y \sec y \tan y d y\\ &=\int u^4 du\\ &=\frac{1}{5}u^5+C\\ &=\frac{1}{5}\sec^5y+C \end{align*}