Back
Problem 4.7.14b
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
(-3/2)csc²x(3x/2)
Problem 7.2.55
Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)
Problem 7.6.23
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
23. y=arcsin(√2t)
Problem 7.3.57
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x), y(0) = 1, y′(0) = 0
Problem 7.3.39
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
Problem 7.3.71
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
71. y = log₂(5θ)
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 7.6.11
Find the values in Exercises 9–12.
11. tan(arcsin(-1/2))
Problem 7.2.6
In Exercises 5 and 6, solve for t.
6. ln(t-2) = ln8 - ln(t)
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 7.1.7
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0
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 7.5.69
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))
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 7.6.17
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
17. lim(x→∞)arcsec(x)
Problem 7.3.111
Evaluate the integrals in Exercises 111–114.
111. ∫₁^(ln x) (1 / t) dt, x > 1
Problem 7.2.56
Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))
Problem 7.3.85
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
85. y = log₂(8t^(ln 2))
Problem 7.8.17
17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.
Problem 7.6.63
Evaluate the integrals in Exercises 53–76.
63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))
Problem 7.3.43
Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt
Problem 7.8.7
7. Order the following functions from slowest growing to fastest growing as x→∞.
a. e^x
b. x^x
c. (ln x)^x
d. e^(x/2)
Problem 7.3.109
Evaluate the integrals in Exercises 97–110.
109. ∫ (dx / (x log₁₀x))
Problem 7.3.83
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)
Problem 7.6.61
Evaluate the integrals in Exercises 53–76.
61. ∫(from 0 to 2)dt/√(8+2t²)
Problem 7.5.32
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
32. lim (x → 0) (3^x - 1) / (2^x - 1)
Problem 7.6.113
Verify the integration formulas in Exercises 111–114.
113. ∫ (arcsin x)² dx = x(arcsin x)² - 2x + 2 √(1 - x²) arcsin x + C
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 7.7.27
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
27. y = (1 - θ)tanh⁻¹(θ)
Problem 7.7.63
Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.
sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞
cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1
tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1
sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1
csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1
coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1
Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.
63. tanh⁻¹(-1/2)