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

∫ (ln x)³/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

arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).

For reference, arcsin(0.6) = 0.64350 to five decimal places.

Evaluate the integrals in Exercises 33–52.

∫ tan⁴(x) sec³(x) dx

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

∫ tan^(-1)(x) / x² dx

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

∫ (x + 3) / (2x³ - 8x) dx

Evaluate the integrals in Exercises 23–32.

∫₀^π √(1 - cos²(θ)) dθ

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

∫ 2 / (x(ln x - 2)³) dx

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

∫ dx / (x² √(4x - 9))

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

∫ 1 / √(x² - 2x + 5) dx

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

∫ θ cos(πθ) dθ

Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year’s rainfall will exceed 17 in.?

Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.

What is the largest value that

∫ from a to b x√(2x - x²) dx

can have for any a and b? Give reasons for your answer.

Evaluate the integrals in Exercises 1–22.

∫ sin⁴(2x) cos(2x) dx

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

∫x e^(3x) dx

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ cos^(-1)(√x) / √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.

∫ (dt / t√(3 + t²)

Average value

In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is

y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.

Find the average value of y over the interval 0 ≤ t ≤ 2π.

In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x² dx) / ((x - 1)(x² + 2x + 1))

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.

∫ (x² / (x² + 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 0 to 1 of (dt / (t - sin t))

(Hint: t ≥ sin t for t ≥ 0)

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

(5x - 7) / (x² - 3x + 2)

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)

To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.

∫ arctan x dx

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

∫ dx / (x √(7 - x²))

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.

∫ (dθ / cos θ - 1)

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

∫ √(1 - (ln x)²) / (x ln x) dx

Evaluate the integrals in Exercises 1–22.

∫₀^π 8cos⁴(2πx) dx

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

∫ x² tan⁻¹(x / 2) dx

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

∫ (x + 2) / (x³ - 2x² - 3x) dx

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.