7. Antiderivatives & Indefinite Integrals

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

∫ dy / (y² − 2y + 2)

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ x^2 / √(x^2 - 4x + 5) dx

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (f) ∫ d𝓍/√36 ―𝓍²

Variations on the substitution method Evaluate the following integrals.                                                                                                        

 ∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍

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

d. ∫(1/eˣ) dx = ln eˣ + C.

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (4/x√(x² - 1))dx

Evaluate the integrals in Exercises 1–14.

∫ √(y² - 25) / y³ dy, where y > 5

71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.

71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

23. ∫ 1/(25 - x²)^(3/2) dx

Evaluate the integrals in Exercises 1–14.

∫ (3 dx) / √(1 + 9x²)

70. Different methods Let I=∫(x+2)/(x+4)dx.

b. Evaluate I without performing long division on the integrand.

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

54. ∫ dx/√(9x² - 25), x > 5/3

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ dx / (1 + x²)

Evaluate the integrals in Exercises 37–46.

∫ √t sin(2t³/²)dt

Evaluate the following integrals.

∫ eˣ/(e²ˣ + 2eˣ + 17) dx