Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ 1/(x(ln(x))²) dx

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).

Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].

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 2 to ∞ of (dx / √(x² - 1))