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 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.

∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)

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

∫ x cos³(x) dx

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ √(x - x²) / x dx

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ x (7x + 5)^(3/2) dx

Evaluate the integrals in Exercises 1–22.

∫₀^(π/6) 3cos⁵(3x) dx

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

∫x e^(3x) dx