In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / x^(1/5)) dx

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (1 - r²)^(5/2) / r⁸ dr

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 ∞ of (dθ / (1 + e^θ))

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

∫ z(ln z)² dz

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.

∫ (dθ / √(2θ - θ²))

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ (√2 - x) / √x 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.

∫ (dt / t√(3 + t²)