The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ 2x e^(−x²) dx

Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.

Evaluate the integrals in Exercises 39–54.

∫ sin(θ) dθ / (cos²θ + cos θ - 2)

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ z(ln z)² dz

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ (√2 - x) / √x dx

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x + 3) / (2x³ - 8x) 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.

∫ (dt / t√(3 + t²)