12. Techniques of Integration

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ cotx·csc³x dx

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 16x^3 (ln(x))^2 dx

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.

∫ θ cos(πθ) dθ

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.

∫x e^(3x) 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 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