7. Antiderivatives & Indefinite Integrals

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (x² / (x⁴ + x²)) dx

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (⁴√x³ + √x⁵) dx

1. Give some examples of analytical methods for evaluating integrals.

4. Is a reduction formula an analytical method or a numerical method? Explain.

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.

18. ∫ dx / (225 − 16x²)

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.

24. ∫ dt / √(1 + 4eᵗ)

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.

31. ∫ √(x² - 8x) dx, x > 8

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.

37. ∫ dx / √(x² + 10x), x >

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.)

Evaluate the following integrals.

∫ eˣ/(e²ˣ + 2eˣ + 17) dx

Evaluate the following integrals.

∫ x/(x² + 6x + 18) dx

6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

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.

11. ∫ 3u / (2u + 7) du

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.

15. ∫ x / √(4x + 1) dx

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

d. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.