12. Techniques of Integration

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

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

∫ [t / (t⁴ − t² − 2)] dt

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 integrals in Exercises 39–54.

∫ 1 / (x⁶(x⁵ + 4)) dx

Evaluate the integral.

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

In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (x³ dx) / (x² - 2x + 1) from -1 to 0

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

∫ [(x + 1) / (x² (x − 1))] dx

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (y + 4) / (y² + y) dy from 1/2 to 1

23-64. Integration Evaluate the following integrals.

29. ∫₋₁² [(5x) / (x² - x - 6)] dx