12. Techniques of Integration

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

∫ e^(-y) cos(y) dy

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

∫ e^(-2x) sin(2x) dx

[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.

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

∫ ln(x + x²) dx

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

∫ z(ln z)² dz

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.

∫ x² ln(x) dx

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

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

∫ x sin(x) cos(x) dx