Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.

Evaluate the integrals in Exercises 1–22.

∫₀^π sin⁵(x/2) dx

In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.

∫ 2y⁴ / (y³ - y² + y - 1) dy

Solve the initial value problems in Exercises 67–70 for x as a function of t.

(t + 1) (dx/dt) = x² + 1 (for t > -1), x(0) = 0

Volume of water in a swimming pool

A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral

V = ∫ from 0 to 50 of 30 · h(x) dx.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ x² tan⁻¹(x / 2) dx

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.

∫ ((2ˣ - 1) / 3ˣ) dx