Question 4 (10 points) Evaluate \( \int \sec ^{2} x \tan ^{5} x d x \)
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To solve the integral \( \int \sec^{2} x \tan^{5} x \, dx \), we can use substitution. Let \( u = \tan x \), which implies that \( du = \sec^{2} x \, dx \). Now, replace \( \tan x \) and \( dx \) in the integral: \[ \int \sec^{2} x \tan^{5} x \, dx = \int u^{5} \, du \] Now, we can integrate: \[ \int u^{5} \, du = \frac{u^{6}}{6} + C \] Substituting back \( u = \tan x \): \[ \frac{\tan^{6} x}{6} + C \] Thus, the answer is: \[ \int \sec^{2} x \tan^{5} x \, dx = \frac{\tan^{6} x}{6} + C \]