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.)

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

88. Evaluate ∫ dx/(2 + cos x).

9–61. Trigonometric integrals Evaluate the following integrals.

38. ∫ tan⁵θ sec⁴θ dθ

79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.

82. ∫ (sin⁻¹(ax)) / x² dx, a > 0

94. The family f(x) = 1/xᵖ revisited Consider the family of functions f(x) = 1/xᵖ, where p is a real number.

For what values of p does the integral ∫(1 to ∞) 1/xᵖ dx exist?

What is its value when it exists?

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.

12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

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

23. ∫ 1/(25 - x²)^(3/2) dx