12. Techniques of Integration

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

12. ∫ (8x + 5)/(2x² + 3x + 1) dx

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

15. x / ((x⁴ - 16)²)

23-64. Integration Evaluate the following integrals.

35. ∫ (x² + 12x - 4)/(x³ - 4x) dx

Evaluate the integrals in Exercises 39–54.

∫ sin(θ) dθ / (cos²θ + cos θ - 2)

7–84. Evaluate the following integrals.

59. ∫ 1/(x⁴ + x²) dx

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

9. 4/(x⁵ - 5x³ + 4x)

Use any method to evaluate the integrals in Exercises 55–66.

∫ (x + 2) / (x³ - 2x² - 3x) dx

23-64. Integration Evaluate the following integrals.

44. ∫₁² 2/[t³(t + 1)] dt

66-68. Areas of regions (Use of Tech) Find the area of the following regions.

66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1

Evaluate the integrals in Exercises 39–54.

∫ 1 / ((x¹/³ - 1)√x) dx

(Hint: Let x = u⁶.)

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x² + x) / (x⁴ - 3x² - 4) dx

Evaluate the integral. 

${\int }_{}^{}\frac{-6x^2+3x+5}{x^3-x}dx$

Use any method to evaluate the integrals in Exercises 55–66.

∫ 2^x / (2²x + 2^x - 2) dx

Express the rational function as a sum or difference or simpler rational expressions.

$\frac{13x+2}{2x^2-x-1}$

23-64. Integration Evaluate the following integrals.

23. ∫ [3 / ((x - 1)(x + 2))] dx