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.

28. ∫ ln² x dx

7–64. Integration review Evaluate the following integrals.

7. ∫ dx / (3 - 5x)^4

23-64. Integration Evaluate the following integrals.

32. ∫ (4x - 2)/(x³ - x) dx

3. Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

9–61. Trigonometric integrals Evaluate the following integrals.

47. ∫ (csc⁴x)/(cot²x) dx

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

23. ∫ 1/(25 - x²)^(3/2) dx

5. What is a reduction formula?

54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).

54. ∫(from 0 to π/2) sin⁶x dx = 5π/32

17-22. Give the partial fraction decomposition for the following expressions.

17. (5x - 7) / (x² - 3x + 2)

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

58. ∫₀^{2π} dt / (4 + 2 sin t)²

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. How would you evaluate ∫ tan¹⁰x sec²x dx?

4. Evaluate ∫ (from 0 to 1) (1/x^(1/5)) dx after writing the integral as a limit.

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

19. ∫ (from 1 to ∞) (3x² + 1)/(x³ + x) dx

7–64. Integration review Evaluate the following integrals.

40. ∫ (1 - x) / (1 - √x) dx

95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.

98. ∫(from 0 to 1) (ln x)/(1+x) dx = -π²/12

92–98. Evaluate the following integrals.

94. ∫ (dt / (t³ + 1))

9–61. Trigonometric integrals Evaluate the following integrals.

59. ∫ from 0 to π/2 of √(1 - cos2x) dx

9–61. Trigonometric integrals Evaluate the following integrals.

60. ∫ from 0 to π/8 of √(1 - cos8x) dx

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.

15. x / ((x⁴ - 16)²)

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

7–64. Integration review Evaluate the following integrals.

60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx

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

26. ∫[√2 to √2] √(x² - 1)/x dx

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

39. ∫ x²/(100 - x²)^(3/2) dx

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

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

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.

34. ∫ dx / (x(x¹⁰ + 1))

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

50. ∫ (from 0 to 9) 1/(x - 1)¹ᐟ³ dx

Evaluate the following integrals.

65. ∫ from 0 to 1/6 1/√(1 - 9x²) dx

9–61. Trigonometric integrals Evaluate the following integrals.

45. ∫ sec²x tan¹ᐟ²x dx

Evaluate the following integrals.

∫ e³ˣ/(eˣ - 1) dx