12. Techniques of Integration

7–84. Evaluate the following integrals.

16. ∫ [1 / (x⁴ – 1)] dx

2. Give an example of each of the following.

b. A repeated linear factor

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

∫ (x + 3) / (2x³ - 8x) dx

73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

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.

6. (4x + 1)/(4x² - 1)

85. Another form of ∫ sec x dx

a. Verify the identity:

sec x = cos x / (1 - sin² x)

b. Use the identity in part (a) to verify that:

∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

Give the partial fraction decomposition for the following expression using strategic substitutions for $x$.

$\frac{4x^2-8x-8}{x(x-2)(x+2)}$

23-64. Integration Evaluate the following integrals.

60.∫ 1/[(y² + 1)(y² + 2)] dy

7–84. Evaluate the following integrals.

45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

23-64. Integration Evaluate the following integrals.

26. ∫₀¹ [1 / (t² - 9)] dt

Evaluate the integral. 

${\int }_0^3\frac{3x+10}{x^2+9x+20}dx$

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

∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ

Use the method of partial fractions to evaluate the integral.

${\int }_{}^{}\frac{5x^3-x^2+7x-2}{(x^2+1)(x^2+2)}dx$

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.

76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ

In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.

∫ 2y⁴ / (y³ - y² + y - 1) dy