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Problem 7.3.112
Evaluate the integrals in Exercises 111–114.
112. ∫₁^(eˣ) (1 / t) dt
Problem 7.6.71
Evaluate the integrals in Exercises 53–76.
71. ∫(from -π/2 to π/2) 2cosθ dθ/(1+(sinθ)²)
Problem 7.2.43
Evaluate the integrals in Exercises 39–56.
43. ∫(from 0 to π)(sin t)/(2 - cos t) dt
Problem 7.7.27
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
27. y = (1 - θ)tanh⁻¹(θ)
Problem 7.3.53
Evaluate the integrals in Exercises 33–54.
53. ∫ (e^r / (1 + e^r)) dr
Problem 7.1.29
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) = 1/x², x > 0
Problem 7.6.117
Solve the initial value problems in Exercises 115–120.
117. dy/dx = 1/(x√(x² - 1)), x > 1; y(2) = π
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 7.1.51
Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).
Problem 7.3.17
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = cos(e^(-θ^2))
Problem 7.3.107
Evaluate the integrals in Exercises 97–110.
107. ∫₀⁹ (2 log₁₀(x + 1) / (x + 1)) dx
Problem 7.3.37
Evaluate the integrals in Exercises 33–54.
∫8e^(x+1) dx
Problem 7.4.25
25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-gluconolactone into gluconic acid, for example,
dy/dt = -0.6y
when t is measured in hours. If there are 100 grams of δ-gluconolactone present when t=0, how many grams will be left after the first hour?
Problem 7.3.143
143.
b. Find the average value of ln(x) over [1, e].
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 7.3.57
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x), y(0) = 1, y′(0) = 0
Problem 7.3.43
Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt
Problem 7.3.39
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
Problem 7.3.117
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.
117. y = (√t)ᵗ
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 7.7.71
Evaluate the integrals in Exercises 31–78.
71. ∫(from √2/3 to 2/3)dy/(|y|√(9y²-1))
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 7.2.10
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
10. y = ln(t^(3/2))
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 7.7.43
Evaluate the integrals in Exercises 41–60.
43. ∫6cosh(x/2 - ln3)dx
Problem 7.5.3
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
3. lim (x → ∞) (5x² - 3x) / (7x² + 1)
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 7.3.45
Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) dx
Problem 7.3.33
Evaluate the integrals in Exercises 33–54.
∫(e^(3x) + 5e^(-x)) dx
Problem 7.5.57
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
57. lim (x → 0⁺) x^(-1/ln x)