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.

∫₋₁¹ (√(1 + x²) sin x) dx

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

∫ (2x + 1) / (x² - 7x + 12) dx

Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.

Evaluate the integrals in Exercises 1–22.

∫ cos³(4x) dx

Evaluate the integrals in Exercises 33–52.

∫ eˣ sec³(eˣ) dx

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

∫ x² sin(x³) dx

Evaluate the integrals in Exercises 39–54.

∫ 1 / ((x¹/³ - 1)√x) dx

(Hint: Let x = u⁶.)

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

∫₀^∞ dx / [(x + 1)(x² + 1)]

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.?

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

∫ (xe^x) / (x + 1)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.

∫ √(9 - w²) dw / w²

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) 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 1 of (dθ / (θ² - 2θ))

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

∫₋∞² (2 dx) / (x² + 4)

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))

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

∫ 2 / (x(ln x - 2)³) dx

Evaluate the integrals in Exercises 1–22.

∫ sin⁴(2x) cos(2x) 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.

∫ (√x / (1 + x³)) dx

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

Evaluate the integrals in Exercises 33–52.

∫ from -π/4 to π/4 of 6 tan⁴(x) dx

Evaluate the integrals in Exercises 39–54.

∫ 1 / (cos θ + sin 2θ) dθ

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

∫ from 0 to 1 x√(1 - x) dx

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

∫ √(x - x²) / x dx

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

∫₀² (s + 1) / √(4 − s²) ds

In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.

∫ 2y⁴ / (y³ - y² + y - 1) dy

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

∫ (x² - 2x + 1) e^(2x) dx

Evaluate the integrals in Exercises 23–32.

∫₀^(π/6) √(1 + sin(x)) dx

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

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

∫ (x + 3) / (2x³ - 8x) 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.

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

∫ 1/(x(ln(x))²) dx