12. Techniques of Integration

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))

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

(t⁴ + 9) / (t⁴ + 9t²)

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

71. ∫ (2x² - 4x)/(x² - 4) dx

7–84. Evaluate the following integrals.

47. ∫ [(2x³ + x² - 2x - 4) / (x² - x - 2)] dx

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

∫ [(x³ + 1) / (x³ − x)] dx

Express the rational function as a sum or difference of two simpler fractions. Use a system of equations.

$\frac{4x^2-21x-40}{x^3-x^2-20x}$

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

82. ∫ [dx / (x√(1 + 2x))]

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).

Evaluate the integral.

${\int }_{}^{}\frac{t^4+t^2-2}{t^3+t}dt$