Evaluate the integrals in Exercises 1–24 using integration by parts.
∫x e^(3x) 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.

∫ dx / (x² √(x² + 1))

Use any method to evaluate the integrals in Exercises 65–70.

∫ x cos³(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) = c * x * √(25 - x²) over [0, 5]

Evaluate the integrals in Exercises 1–22.

∫₀^(π/6) 3cos⁵(3x) dx

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.?

In Exercises 67–73, use integration by parts to establish the reduction formula.

∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx