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

∫₋∞^∞ (x dx) / (x² + 4)^(3/2)

Evaluate the integrals in Exercises 53–58.

∫ sin(2x) cos(3x) dx

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3

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

∫₁^∞ dx / [x√(x² − 1)]

Evaluate the integrals in Exercises 33–52.

∫ sec(x) tan²(x) dx

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)

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)))

Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.

Evaluate the integrals in Exercises 1–14.

∫ √(1 - 9t²) dt

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ dx / (4 - x²)^(3/2) from 0 to 1

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

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ (sin⁻¹ x)² / √(1 - x²) dx

Expand the quotients in Exercises 1–8 by partial fractions.

(5x - 7) / (x² - 3x + 2)

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

∫₀¹ (−ln(x)) dx

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

∫ 2 sin^2(t) sec^4(t) dt

Evaluate the integrals in Exercises 33–52.

∫ sec⁶(x) dx

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.

∫ e^(z + eᶻ) dz

Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

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

∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)

Evaluate the integrals in Exercises 53–58.

∫ from -π/2 to π/2 of cos(x) cos(7x) dx

Evaluate the integrals in Exercises 53–58.

∫ from 0 to π/2 of sin(x) cos(x) dx

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

∫ x arctan(x) dx

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.

∫ (2 ln(z³)) / (16z) dz

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)

Evaluate the integrals in Exercises 1–14.

∫ (3 dx) / √(1 + 9x²)

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.

∫ sin(t / 3) sin(t / 6) dt

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.

∫ (dx / ((2x + 1)√(4x + 4x²)))

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ z(ln z)² dz

Use any method to evaluate the integrals in Exercises 55–66.

∫ 2 / (x(ln x - 2)³) dx