Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (xe^x) / (x + 1)² dx

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.

∫ dx / (x - √x)

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 2 of (dx / (1 - x))

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ csc³(√θ) / √θ dθ

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Use numerical integration to estimate the value of

π = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.

Evaluate the integrals in Exercises 1–22.

∫₀^(π/2) sin²(x) dx

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ dx / (x² + 2x)