12. Techniques of Integration

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

∫ [1 / √(e^s + 1)] ds

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

∫ 9 dv / (81 − v⁴)

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

∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx

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)

88. Evaluate ∫ dx/(2 + cos x).

23-64. Integration Evaluate the following integrals.

54. ∫ (z + 1)/[z(z² + 4)] dz

85. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. More than one integration method can be used to evaluate ∫ (1 / (1 - x²)) dx.

23-64. Integration Evaluate the following integrals.

41. ∫₋₁¹ x/(x + 3)² dx

23-64. Integration Evaluate the following integrals.

57. ∫ (x³ + 5x)/(x² + 3)² dx

23-64. Integration Evaluate the following integrals.

32. ∫ (4x - 2)/(x³ - x) dx

Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.

∫ (1 + tan x) sec²x dx

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

$\frac{4}{x^3-3x^2-x+3}$

93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:

vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).

d. Which car ultimately gains the lead and remains in front?

65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.

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

(5x - 7) / (x² - 3x + 2)

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

∫ (8x² + 8x + 2) / (4x² + 1)² dx