📘 integral calculus

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:

1. **Problem:** Evaluate $$\int \sqrt{x} (2\sqrt{x^3} - 1)^3 \, dx$$ by (a) expanding and (b) substitution. 2. **Step 1: Express the integrand clearly.**

1. **Problem:** Evaluate the integral $$\int_0^{\frac{\pi}{2}} \cos^2 x \sin^6 x \, dx$$. 2. **Formula and rules:** We use the Beta function relation for integrals of the form $$\i

1. **State the problem:** We are given two curves: $x=2\sqrt{y}$ and $y=2\sqrt{x}$.