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

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

∫ (√2 - x) / √x dx

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₁^∞ (1 / x^(1/5)) dx

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

∫x e^(3x) dx

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

∫ z(ln z)² dz

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)

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

∫ (x + 3) / (2x³ - 8x) dx

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.

∫ √(9 - w²) dw / w²

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

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.?

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

∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

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 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