Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 2 sin^2(t) sec^4(t) dt

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)

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.

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.

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

∫ tan^(-1)(x) / x² dx