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

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

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?

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)

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.

7–84. Evaluate the following integrals.

56. ∫ from π to 3π/2 sin2x e^(sin²x) 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))

74. A secant reduction formula

Prove that for positive integers n ≠ 1,

∫ secⁿ x dx = (secⁿ⁻² x tan x)/(n − 1) + (n − 2)/(n − 1) ∫ secⁿ⁻² x dx.

(Hint: Integrate by parts with u = secⁿ⁻² x and dv = sec² x dx.)

9–61. Trigonometric integrals Evaluate the following integrals.

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

Evaluate the following integrals.

65. ∫ from 0 to 1/6 1/√(1 - 9x²) dx

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

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

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

7–64. Integration review Evaluate the following integrals.

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

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

50. ∫ (from 0 to 9) 1/(x - 1)¹ᐟ³ dx

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

31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv

9–61. Trigonometric integrals Evaluate the following integrals.

45. ∫ sec²x tan¹ᐟ²x dx

Evaluate the following integrals.

∫ e³ˣ/(eˣ - 1) dx