Evaluate the integrals in Exercises 1–22.
∫₀^π 8 sin⁴(y) cos²(y) dy

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3

Evaluate the integrals in Exercises 1–22.

∫ 7cos⁷(t) dt

Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 0 to π/2 of (cot θ dθ)

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (√x / (1 + x³)) dx

Hint: Let u = x^(3/2).

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ (4r dr) / √(1 − r⁴)