Evaluate the limits in Exercise 7 and 8.
lim (x → ∞) ∫₋ˣ^ˣ sin t dt

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x² + x) / (x⁴ - 3x² - 4) dx

Finding arc length

Find the length of the curve

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

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 integrals in Exercises 1–6.

∫ x arcsin x dx

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.

∫ cos²(2θ) sin(θ) dθ

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)