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Problem 7.6.84
Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 to 4)2dx/(x²-6x+10)
Problem 7.2.39
Evaluate the integrals in Exercises 39–56.
39. ∫(from -3 to -2)dx/x
Problem 7.7.33
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
33. y = csch⁻¹(1/2)^θ
Problem 7.3.143
143.
b. Find the average value of ln(x) over [1, e].
Problem 7.3.81
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
81. y = log₁₀(e^x)
Problem 7.1.9
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 1 - x/2, x ≤ 0
x/(x + 2), x > 0
Problem 7.7.43
Evaluate the integrals in Exercises 41–60.
43. ∫6cosh(x/2 - ln3)dx
Problem 7.3.11
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = xe^x-e^x
Problem 7.6.13
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
13. lim(x → 1⁻)arcsin(x)
Problem 7.4.10
Solve the differential equation in Exercises 9–22.
10. (dy/dx) = x²√y, y > 0
Problem 7.3.77
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
77. y = log₃(((x + 1)/(x − 1))^(ln 3))
Problem 7.4.47
47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?
Problem 7.3.95
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
Problem 7.5.63
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
63. lim (x → ∞) ((x + 2)/(x - 1))^x
Problem 7.6.9
Find the values in Exercises 9–12.
9. sin(arccos((√2)/2))
Problem 7.6.102
Evaluate the integrals in Exercises 91–102.
102. ∫(from -1/3 to 1/√3)(cos(arctan 3x))/(1+9x²) dx
Problem 7.5.41
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
41. lim (x → 0⁺) (ln x)² / ln(sin x)
Problem 7.3.67
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
Problem 7.7.59
Evaluate the integrals in Exercises 41–60.
59. ∫(from -ln2 to 0)cosh²(x/2) dx
Problem 7.1.1
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Problem 7.7.9
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
9. (sinh(x)+cosh(x))⁴
Problem 7.3.23
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(cost+lnt)
Problem 7.5.51
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)
Problem 7.2.64
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
64. y = 1/(t(t+1)(t+2))
Problem 7.2.61
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
61. y = √(θ + 3) sin θ
Problem 7.5.35
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
35. lim (x → 0⁺) ln(x² + 2x) / ln x
Problem 7.7.57
Evaluate the integrals in Exercises 41–60.
57. ∫(from 1 to 2)cosh(ln t)/t dt
Problem 7.6.93
Evaluate the integrals in Exercises 91–102.
93. ∫(arcsin x)²dx/√(1-x²)
Problem 7.2.29
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
29. y = ln(1/(x√(x+1)))
Problem 7.4.21
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
