Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ csc³(√θ) / √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.

∫ dx / (x - √x)

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

∫ (xe^x) / (x + 1)² dx

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))

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

∫ dx / (x² + 2x)

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

∫ x⁵ e³ˣ dx

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ ln(x + x²) dx