Evaluate the integrals in Exercises 1–22.
∫ cos³(2x) sin⁵(2x) dx

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)

Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.

∫ csc θ dθ

Evaluate the integrals in Exercises 1–22.

∫ sin⁴(2x) cos(2x) dx

Evaluate the integrals in Exercises 1–22.

∫ cos³(4x) dx

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

∫ x² sin(x³) dx

Evaluate the integrals in Exercises 33–52.

∫ eˣ sec³(eˣ) dx

Evaluate the integrals in Exercises 33–52.

∫ tan⁴(x) sec³(x) dx

Evaluate the integrals in Exercises 33–52.

∫ sec⁶(x) dx