Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (ln x)³/x dx

Evaluate the integrals in Exercises 53–58.

∫ from 0 to π/2 of sin(x) cos(x) 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.

∫ (x² dx) / (x² - 1)^(5/2), where x > 1

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]

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

∫₀^π/2 x³ cos 2x dx

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

∫ x² sin(x³) dx

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

∫₀^∞ dx / [(x + 1)(x² + 1)]