12. Techniques of Integration

Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.

lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ x² sin(x) dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ (x² - 2x + 1) e^(2x) dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ arcsin(y) dy

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ 4x sec²(2x) dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ (r² + r + 1) e^r dr

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ t² e^(4t) dt

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

∫ x⁵ e³ˣ dx

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

∫ √x e√x dx

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

∫₀^π/2 x³ cos 2x dx

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

∫₀¹/√2 2x arcsin(x²) dx

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

∫ x² tan⁻¹(x / 2) dx

Evaluate ∫ x³ √(1 - x²) dx using:

a. Integration by parts.

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ arccos(x / 2) dx

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ x² sin(1 − x) dx