1. **Problem:** Evaluate the integral $$\int \sec^5 x \, dx$$. 2. **Formula and rules:** We use the reduction formula for powers of secant from question 3a: $$\int \sec^n x \, dx = \frac{\sec^{n-2} x \tan x}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx$$ This formula helps reduce the power of secant step-by-step. 3. **Apply the formula for $n=5$:** $$\int \sec^5 x \, dx = \frac{\sec^{3} x \tan x}{4} + \frac{3}{4} \int \sec^{3} x \, dx$$ 4. **Next, evaluate $\int \sec^3 x \, dx$ using the same formula with $n=3$:** $$\int \sec^3 x \, dx = \frac{\sec x \tan x}{2} + \frac{1}{2} \int \sec x \, dx$$ 5. **Recall the integral of $\sec x$:** $$\int \sec x \, dx = \ln |\sec x + \tan x| + C$$ 6. **Substitute back:** $$\int \sec^3 x \, dx = \frac{\sec x \tan x}{2} + \frac{1}{2} \ln |\sec x + \tan x| + C$$ 7. **Substitute into step 3:** $$\int \sec^5 x \, dx = \frac{\sec^{3} x \tan x}{4} + \frac{3}{4} \left( \frac{\sec x \tan x}{2} + \frac{1}{2} \ln |\sec x + \tan x| \right) + C$$ 8. **Simplify:** $$\int \sec^5 x \, dx = \frac{\sec^{3} x \tan x}{4} + \frac{3 \sec x \tan x}{8} + \frac{3}{8} \ln |\sec x + \tan x| + C$$ **Final answer:** $$\boxed{\int \sec^5 x \, dx = \frac{\sec^{3} x \tan x}{4} + \frac{3 \sec x \tan x}{8} + \frac{3}{8} \ln |\sec x + \tan x| + C}$$