The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ

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

∫₋∞^∞ (x dx) / (x² + 4)^(3/2)

Evaluate the integrals in Exercises 1–14.

∫ (2 dx) / (x³ √(x² - 1)), where x > 1

Evaluate the integrals in Exercises 39–54.

∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt

Evaluate the integrals in Exercises 33–52.

∫ sec(x) tan²(x) 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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))

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))