Evaluate the indefinite integral. (Use $C$ for the constant of integration.)
\[
\int \sec ^{2}(\theta) \tan ^{8}(\theta) d \theta
\]

Answer

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Answer

Final Answer: $\boxed{\frac{\tan ^{9}(\theta)}{9}+C}$

Steps

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: $\boxed{\frac{\tan ^{9}(\theta)}{9}+C}$