Problem
Evaluate the indefinite integral. (Use $C$ for the constant of integration.) \[ \int \sec ^{2}(\theta) \tan ^{8}(\theta) d \theta \]
Solution
Step 1 ：Let \(u = \tan(\theta)\), then \(du = \sec^2(\theta) d\theta\).
Step 2 ：The integral then becomes \(\int u^8 du\).
Step 3 ：Solve the integral to get \(u^9/9\).
Step 4 ：Substitute \(\tan(\theta)\) back in for \(u\) to get the final answer.
Step 5 ：Final Answer: \(\boxed{\frac{\tan ^{9}(\theta)}{9}+C}\)
