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)

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ dr / r^0.999

In Exercises 67–73, use integration by parts to establish the reduction formula.

∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) 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)