Use any method to evaluate the integrals in Exercises 55–66.
∫ x² √(1 - x²) dx

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ dy / (y√(1 + (ln y)²)) from 1 to e

Expand the quotients in Exercises 1–8 by partial fractions.

(5x - 7) / (x² - 3x + 2)

Average value

In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is

y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.

Find the average value of y over the interval 0 ≤ t ≤ 2π.

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 2 sin^2(t) sec^4(t) dt

Evaluate ∫ x³ √(1 - x²) dx using:

c. A trigonometric substitution.

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 π to ∞ of ((1 + sin x) / x² dx)