Evaluate the integrals in Exercises 1–22.
∫ cos³(4x) 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]

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

∫₀^π/2 x³ cos 2x 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 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.

∫ (2x + 1) / (x² - 7x + 12) dx

Average Value: Find the average value of the function f(x) = 1 / (1 - sin θ) on the interval [0, π/6].

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

∫₀^∞ dx / [(x + 1)(x² + 1)]