7. Antiderivatives & Indefinite Integrals

7–64. Integration review Evaluate the following integrals.

62. ∫ (-x⁵ - x⁴ - 2x³ + 4x + 3) / (x² + x + 1) dx

68. Different methods

b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.

92–98. Evaluate the following integrals.

94. ∫ (dt / (t³ + 1))

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

∫ (2x +1)² dx

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

∫ ((1/x²) - (2/(x⁵⸍²))) dx

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

∫ (12/x)dx

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.)