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Problem 5.5.63
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₁/₃^¹/√³ 4/(9𝓍² + 1) d𝓍
Problem 5.5.66
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₀ᵉ² (ln p)/p dp
Problem 5.5.70
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₋₁¹ (𝓍―1) (𝓍²―2𝓍)⁷ d𝓍
Problem 5.5.74
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₀^π/⁴ eˢᶦⁿ² ˣ sin 2𝓍 d𝓍
Problem 5.5.79
Variations on the substitution method Evaluate the following integrals.
∫ 𝓍/(√𝓍―4) d𝓍
Problem 5.5.80
Variations on the substitution method Evaluate the following integrals.
∫ y²/(y + 1)⁴ dy
Problem 5.5.88
Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.
∫ sin² 𝓍 d𝓍
Problem 5.5.90
Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.
∫₀^π/⁴ cos² 8θ dθ
Problem 5.5.92
Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.
∫ 𝓍 cos²𝓍² d𝓍
Problem 5.5.95a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.
(a) ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍 = ½ (ƒ(𝓍))² + C.
Problem 5.5.95b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.
(b) ∫ (ƒ(𝓍))ⁿ ƒ'(𝓍) d𝓍 = 1/(n + 1) (ƒ(𝓍))ⁿ⁺¹ + C , n ≠ ―1 .
Problem 5.5.95c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.
(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .
Problem 5.5.100a
Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.
(a) ∫¹₋₁ 𝓍ƒ(𝓍²) d𝓍
Problem 5.5.100b
Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.
(b) ∫²₋₂ 𝓍²ƒ(𝓍³) d𝓍
Problem 5.5.87
Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.
∫₋π^π cos² 𝓍 d𝓍
Problem 5.5.116
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
∫ 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)
Problem 5.5.117
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
∫ d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)
Problem 5.5.119
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)
Problem 5.5.81
Variations on the substitution method Evaluate the following integrals.
∫ 𝓍/(∛𝓍 + 4) d𝓍
Problem 5.5.82
Variations on the substitution method Evaluate the following integrals.
∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍
Problem 5.5.84
Variations on the substitution method Evaluate the following integrals.
∫ (𝒵 + 1) √(3𝒵 + 2) d𝒵
Problem 5.5.98
Areas of regions Find the area of the following regions.
The region bounded by the graph of ƒ(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and 𝓍= 6
Problem 5.5.96
Areas of regions Find the area of the following regions.
The region bounded by the graph of ƒ(𝓍) = x /√(𝓍² ―9) and the 𝓍-axis between and 𝓍 = 4 and 𝓍= 5
Problem 5.5.1
On which derivative rule is the Substitution Rule based?
Problem 5.5.3
The composite function ƒ(g(𝓍)) consists of an inner function g and an outer function ƒ. If an integrand includes ƒ(g(𝓍)), which function is often a likely choice for a new variable u?
Problem 5.5.5
When using a change of variables u = g(𝓍) to evaluate the definite integral ∫ₐᵇ ƒ(g(𝓍)) g' (𝓍) d(𝓍), how are the limits of integration transformed?
Problem 5.5.102
Average distance on a triangle Consider the right triangle with vertices (0,0) ,(0,b) , and (a,0) , where a > 0 and b > 0. Show that the average vertical distance from points on the 𝓍-axis to the hypotenuse is b/2 , for all a > 0 .
Problem 5.5.106
General results Evaluate the following integrals in which the function ƒ is unspecified. Note that ƒ⁽ᵖ⁾ is the pth derivative of ƒ and ƒᵖ is the pth power of ƒ. Assume ƒ and its derivatives are continuous for all real numbers.
∫ (5 ƒ³ (𝓍) + 7ƒ² (𝓍) + ƒ (𝓍 )) ƒ'(𝓍) d𝓍