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Problem 7.3.95
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
Problem 7.2.41
Evaluate the integrals in Exercises 39–56.
41. ∫2y dy/(y²-25)
Problem 7.4.41
41. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room where the temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions.
a. How much longer would it take the soup to cool to 35°C?
Problem 7.6.69
Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))
Problem 7.3.53
Evaluate the integrals in Exercises 33–54.
53. ∫ (e^r / (1 + e^r)) dr
Problem 7.3.73
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²
Problem 7.6.17
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
17. lim(x→∞)arcsec(x)
Problem 7.2.87
Solve the initial value problems in Exercises 87 and 88.
87. dy/dx = 1 + 1/x, y(1) = 3
Problem 7.1.35
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = (x + b) / (x − 2), b > −2 and constant
Problem 7.2.56
Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))
Problem 7.3.139
In Exercises 139–142, find the length of each curve.
139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.
Problem 7.2.55
Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)
Problem 7.2.47
Evaluate the integrals in Exercises 39–56.
47. ∫(from 2 to 4)dx/(x(ln x)²)
Problem 7.5.30
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
30. lim (θ → 0) ((1/2)^θ - 1) / θ
Problem 7.3.79
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
79. y = θ sin(log₇ θ)
Problem 7.3.103
Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln 2 · log₂x / x) dx
Problem 7.2.14
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
14. y = ln(2θ+2)
Problem 7.3.49
Evaluate the integrals in Exercises 33–54.
49. ∫ e^(sec πt) sec πt tan πt dt
Problem 7.6.23
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
23. y=arcsin(√2t)
Problem 7.2.35
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
35. y = ln((x²+1)^5/√(1-x))
Problem 7.5.37
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
Problem 7.6.130
130. Use the identity arccot(u)=π/2 - arctan(u) to derive the formula for the derivative of arccot(u) in Table 7.4 from the formula for the derivative of arctan(u).
Problem 7.2.16
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
16. y = (ln x)³
Problem 7.1.27
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x³ + 1
Problem 7.6.77
Evaluate the integrals in Exercises 77–90.
77. ∫dx/√(-x²+4x-3)
Problem 7.2.66
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))
Problem 7.3.122
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)
Problem 7.3.5
5. e^(2t)-3e^t = 0
Problem 7.7.31
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
31. y = cos⁻¹(x) - x sech⁻¹(x)
Problem 7.4.21
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)