118. Two worthy integrals

b. Let f be any positive continuous function on the interval [0, π/2]. Evaluate

∫ from 0 to π/2 of [f(cos x) / (f(cos x) + f(sin x))] dx.

(Hint: Use the identity cos(π/2 − x) = sin x.)

(Source: Mathematics Magazine 81, 2, Apr 2008)

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.

a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

71. ∫ (2x² - 4x)/(x² - 4) dx

95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(​n​) for the given values of n.

96. ∫ (from 1 to 3) dx/(x³ + x + 1); n = 4

95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(​n​) for the given values of n.

97. ∫ (from 0 to 1) tan(x²) dx; n = 40

109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given by

v(t) = 10 * sin(3t).

123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].

122. Comparing areas The region R₁ is bounded by the graph of y = tan(x) and the x-axis on the interval [0, π/3].

The region R₂ is bounded by the graph of y = sec(x) and the x-axis on the interval [0, π/6]. Which region has the greater area?

106. Arc length Find the length of the curve y = (x / 2) * sqrt(3 - x^2) + (3 / 2) * sin^(-1)(x / sqrt(3)) from x = 0 to x = 1.

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

82. ∫ (from -∞ to -1) dx/(x - 1)⁴

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

84. ∫ (from 0 to π) sec²x dx*(Note: Potential improperness at x = π/2)*

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

86. ∫ (from -∞ to ∞) x³/(1 + x⁸) dx

89–91. Comparison Test Determine whether the following integrals converge or diverge.

89. ∫ (from 1 to ∞) dx/(x⁵ + x⁴ + x³ + 1)

92. Integral with a parameter For what values of p does the integral

∫ (from 1 to ∞) dx/xlnᵖ(x) converge, and what is its value (in terms of p)?

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

43. ∫ eˣ sin(x) dx

102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.

102. About the y-axis

102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.

104. About the line y = 1

110. Comparing distances Suppose two cars started at the same time and place (t = 0 and s = 0). The velocity of car A (in mi/hr) is given by

u(t) = 40 / (t + 1) and the velocity of car B (in mi/hr) is given by v(t) = 40 * e^(-t/2).

b. After t = 3 hr, which car has traveled farther?

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

12. ∫ (8x + 5)/(2x² + 3x + 1) dx

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

63. ∫ dx/(x² - 2x - 15)

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

65. ∫ (from 0 to 1) dy/((y + 1)(y² + 1))

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

38. ∫ (from π/4 to π/2) x csc²x dx

119. {Use of Tech} Comparing volumes Let R be the region bounded by y = ln(x), the x-axis, and the line x = a, where a > 1.

b. Find the volume V₂(a) of the solid generated when R is revolved about the y-axis (as a function of a).

120. Equal volumes

a. Let R be the region bounded by the graph of f(x) = x^(-p) and the x-axis, for x ≥ 1. Let V₁ and V₂ be the volumes of the solids generated when R is revolved about the x-axis and the y-axis, respectively, if they exist. For what values of p (if any) is V₁ = V₂?

b. Repeat part (a) on the interval [0, 1].

101. Comparing volumes Let R be the region bounded by the graph of y = sin(x) and the x-axis on the interval [0, π]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or about the y-axis?

71. Different Methods

Let I = ∫ (x²)/(x + 1) dx.

b. Evaluate I by first performing long division on the integrand.

7–64. Integration review Evaluate the following integrals.

10. ∫ e^(3 - 4x) dx

7–64. Integration review Evaluate the following integrals.

7. ∫ dx / (3 - 5x)^4

7–64. Integration review Evaluate the following integrals.

8. ∫ (9x - 2)^(-3) dx

7–64. Integration review Evaluate the following integrals.

12. ∫ from -5 to 0 of dx / √(4 - x)