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 / (θ² - 1))

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ e^(-2x) sin(2x) dx

In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x³ dx) / (x² - 2x + 1) from -1 to 0

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ √(x² - 4) / x dx

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 16x^3 (ln(x))^2 dx

The length of one arch of the curve y = sin x is given by

L = ∫(from 0 to π) √(1 + cos²(x)) dx.

Estimate L by Simpson's Rule with n = 8.

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

∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)