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

∫ z(ln z)² dz

[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

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 / (x - √x)

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

∫ csc³(√θ) / √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 0 to ∞ of (dθ / (1 + e^θ))

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 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

Evaluate the integrals in Exercises 33–52.

∫ cot⁶(2x) dx

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

∫ 2^x / (2²x + 2^x - 2) 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.

∫ (dθ / √(2θ - θ²))

Evaluate the integrals in Exercises 33–52.

∫ from -π/4 to π/4 of 6 tan⁴(x) dx

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

∫ dx / (x² + 2x)

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

∫ √(x - 2) / √(x - 1) dx

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

∫ (x² dx) / ((x - 1)(x² + 2x + 1))

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

∫ x cos³(x) dx

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

∫₀^∞ dx / [(x + 1)(x² + 1)]

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

∫ √(x - x²) / x dx

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

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

Evaluate the integrals in Exercises 39–54.

∫ 1 / (cos θ + sin 2θ) dθ

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)

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = (1/x) over [c, c + 1]

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

∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx

Average Value: Find the average value of the function f(x) = 1 / (1 - sin θ) on the interval [0, π/6].

Evaluate the integrals in Exercises 1–22.

∫₀^(π/2) sin²(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.

Evaluate the integrals in Exercises 1–22.

∫₀^(π/6) 3cos⁵(3x) dx

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

∫₋₈¹ dx / x^(1/3)

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

∫ x² sin(x³) dx

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

∫ (x + 2) / (x³ - 2x² - 3x) dx