Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.

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

∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)

Evaluate the integrals in Exercises 53–58.

∫ from 0 to π/2 of sin(x) cos(x) 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.

∫ 3 sinh(x/2 + ln 5) dx

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

∫₋∞² (2 dx) / (x² + 4)

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

∫₋₈¹ dx / x^(1/3)

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

∫ from 0 to 1 x√(1 - x) dx