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.
∫ (6 dy / √y(1 + y))

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

∫₀^∞ 2e^(−θ) sinθ dθ

Evaluate the integrals in Exercises 33–52.

∫ sec(x) tan²(x) dx

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ dy / (y√(1 + (ln y)²)) from 1 to e

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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))

Evaluate ∫ x³ √(1 - x²) dx using:

c. A trigonometric substitution.

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 π to ∞ of ((1 + sin x) / x² dx)