Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫₀^π/2 x³ cos 2x 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

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

∫ (ln x)³/x dx

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]

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.

∫ cos³(4x) dx

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

∫₀^∞ dx / [(x + 1)(x² + 1)]