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.

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

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

(5x - 7) / (x² - 3x + 2)

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

∫ 2 sin^2(t) sec^4(t) dt

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

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Use numerical integration to estimate the value of

arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).

For reference, arcsin(0.6) = 0.64350 to five decimal places.