12. Techniques of Integration

23-64. Integration Evaluate the following integrals.

38. ∫₀⁵ 2/(x² - 4x - 32) dx

Evaluate the integrals in Exercises 33–36.

∫ [1 / (x(9 - x²))] dx

Expand the quotients in Exercises 1–8 by partial fractions.

(2x + 2) / (x² - 2x + 1)

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

∫ (s⁴ + 81) / (s(s² + 9)²) ds

Expand the quotients in Exercises 1–8 by partial fractions.

z / (z³ - z² - 6z)

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

∫ (z + 1) / [z²(z² + 4)] dz

Evaluate the integral. 

${\int }_{}^{}\frac{4x^2+2x+3}{x^2+x}dx$

17-22. Give the partial fraction decomposition for the following expressions.

20. (x² - 4x + 11) / ((x - 3)(x - 1)(x + 1))

Express the integrand as a sum of partial fractions and evaluate the integral.

${\int }_{}^{}\frac{2}{x^3+x^2-x-1}dx$

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [x / (x² + 4x + 3)] dx

2. Give an example of each of the following.

d. A repeated irreducible quadratic factor

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

∫ e^t dt / (e^(2t) + 3e^t + 2)

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

65. ∫ (from 0 to 1) dy/((y + 1)(y² + 1))

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

34. ∫ dx / (x(x¹⁰ + 1))

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.

12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))