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
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)
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
