Expand the quotients in Exercises 1–8 by partial fractions.
(5x - 7) / (x² - 3x + 2)

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

Use any method to evaluate the integrals in Exercises 55–66.

∫ x² √(1 - x²) dx

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.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ e^(z + eᶻ) dz