Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫(from π/2 to 2π/3) cos θ dθ / (sin θ cos θ + sin θ)

Finding arc length

Find the length of the curve

y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:

a. the y-axis.

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

∫ (cos(√x))/(√x) dx

Length of a curve

Find the length of the curve

y = ∫(from 1 to x) sqrt(sqrt(t) - 1) dt, where 1 ≤ x ≤ 16.

Evaluate the limits in Exercise 7 and 8.

lim (x → ∞) ∫₋ˣ^ˣ sin t dt

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ cos t dt / (1 - cos t)