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

∫₀² dx / √|x − 1|

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

∫₀¹ (4r dr) / √(1 − r⁴)

Evaluate the integrals in Exercises 33–52.

∫ eˣ sec³(eˣ) dx

Use any method to evaluate the integrals in Exercises 65–70.

∫ x cos³(x) dx

Evaluate the integrals in Exercises 39–54.

∫ 1 / ((x¹/³ - 1)√x) dx

(Hint: Let x = u⁶.)

Evaluate the integrals in Exercises 1–22.

∫₀^(π/2) sin²(x) dx

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

∫ p⁴ e^(-p) dp

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ dx / (x² + 2x)

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

∫ (x² + x) / (x⁴ - 3x² - 4) 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

Evaluate the integrals in Exercises 1–22.

∫ sin⁴(2x) cos(2x) dx

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

∫ √(9 - w²) dw / w²

Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.

86. Find the volume of the solid generated by revolving the region about the x-axis.

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

Evaluate the integrals in Exercises 1–14.

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

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

∫ (x dx) / (25 + 4x²)

In Exercises 67–73, use integration by parts to establish the reduction formula.

∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx

Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.

∫ cos²(2θ) sin(θ) dθ

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 1 to ∞ of ((1 / (e^x - 2^x)) dx)

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

∫ √(x - 2) / √(x - 1) dx

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

∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)

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

∫₋∞^∞ (x dx) / (x² + 4)^(3/2)

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

∫ x² √(1 - x²) 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.

∫ (6 dy / √y(1 + y))

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.

∫₋₁³ (4x² - 7) / (2x + 3) dx

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.

∫ sin(θ) sin(2θ) sin(3θ) dθ

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

∫ e^(-3t) sin(4t) dt

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

∫ (cos(√x))/(√x) dx

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

∫ (ln x)³/x dx