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 √(x² - 4) dx

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

∫ 1 / √(x² - 2x + 5) dx

Evaluate the integrals in Exercises 33–52.

∫ tan⁴(x) sec³(x) 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 1–24 using integration by parts.

∫x e^(3x) dx

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

∫ (x + 2) / (x³ - 2x² - 3x) 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))

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)

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.

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

∫ (2x + 1) / (x² - 7x + 12) 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.

∫ dx / (x - √x)

Evaluate the integrals in Exercises 1–22.

∫ cos³(4x) 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)

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

∫₀^∞ dx / [(x + 1)(x² + 1)]

Evaluate the integrals in Exercises 23–32.

∫₀^(π/6) √(1 + sin(x)) dx

(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))

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 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (x dx) / (25 + 4x²)

Evaluate the integrals in Exercises 33–52.

∫ cot⁶(2x) dx

Evaluate the integrals in Exercises 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

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

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

∫ (cos(√x))/(√x) dx

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

∫₋∞² (2 dx) / (x² + 4)

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

∫ 2^x / (2²x + 2^x - 2) dx

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

∫ from 0 to 1 x√(1 - x) dx

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

∫ cos²(2θ) sin(θ) dθ

[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 = sin x, 0 ≤ x ≤ π

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

∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

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

∫₀¹ (4r dr) / √(1 − r⁴)

Evaluate the integrals in Exercises 39–54.

∫ 1 / (cos θ + sin 2θ) dθ

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

∫ (ln x)³/x dx