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)

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))

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.

Solve the initial value problems in Exercises 53–56 for y as a function of x.

(x² + 1)² (dy/dx) = √(x² + 1), where y(0) = 1

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

∫ 8 cot^4(t) dt