Question

Evaluate: \[ \int \tan^4 x \sec^2 x \, dx. \]

Hint:

When dealing with integrals involving \( \sec^2 x \), consider using the substitution \( u = \tan x \).

Answer 1

We are asked to evaluate the integral: \[ \int \tan^4 x \sec^2 x \, dx. \]

Step 1: Use substitution. We use the substitution \( u = \tan x \), which implies \( du = \sec^2 x \, dx \). Substitute into the integral: \[ \int \tan^4 x \sec^2 x \, dx = \int u^4 \, du. \]

Step 2: Integrate. Now, integrate \( u^4 \): \[ \int u^4 \, du = \frac{u^5}{5} + C. \]

Step 3: Substitute back. Now, substitute \( u = \tan x \) back into the result: \[ \frac{u^5}{5} + C = \frac{\tan^5 x}{5} + C. \] Conclusion: The value of the integral is: \[ \boxed{\frac{\tan^5 x}{5} + C}. \]