Evaluate the integrals in Exercises 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

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 1 of (dt / (t - sin t))

(Hint: t ≥ sin t for t ≥ 0)

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

∫ (xe^x) / (x + 1)² dx

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

Evaluate the integrals in Exercises 33–52.

∫ from -π/4 to π/4 of 6 tan⁴(x) dx

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.

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

∫ x cos³(x) dx

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Use numerical integration to estimate the value of

π = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.

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

∫ √(x - x²) / x dx

Evaluate the integrals in Exercises 1–14.

∫ 5 dx / √(25x² - 9), where x > 3/5

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

∫ x (7x + 5)^(3/2) dx

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 2 to ∞ of (dx / √(x² - 1))

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)

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

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.

y = x²/4, 0 ≤ x ≤ 2

Evaluate the integrals in Exercises 39–54.

∫ 1 / (cos θ + sin 2θ) dθ

Evaluate the integrals in Exercises 53–58.

∫ from 0 to π/2 of sin(x) cos(x) dx

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

∫ x² sin(x³) dx

Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.

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 31–56. Some integrals do not require integration by parts.

∫₀^π/2 x³ cos 2x dx

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

∫ 1/(x(ln(x))²) 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.

∫ (1 - r²)^(5/2) / r⁸ dr

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.

∫ (√x / (1 + x³)) dx

Hint: Let u = x^(3/2).

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.

∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

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)

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

∫₀¹ (4r dr) / √(1 − r⁴)

Evaluate the integrals in Exercises 53–58.

∫ sin(2x) cos(3x) dx

Evaluate the integrals in Exercises 1–22.

∫₀^(π/2) sin²(x) dx

Evaluate the integrals in Exercises 33–52.

∫ cot⁶(2x) dx