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)

Solve the differential equation in Exercises 9–22.

21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)

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?

Evaluate the integrals in Exercises 39–56.

55. ∫dx/(2√x + 2x)

In Exercises 7–10, determine from its graph if the function is one-to-one.

f(x) = 3 - x, x < 0

      = 3, x ≥ 0

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

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))

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))

Evaluate the integrals in Exercises 33–54.

∫(from ln4 to ln9)e^(x/2)dx

Evaluate the integrals in Exercises 111–114.

111. ∫₁^(ln x) (1 / t) dt, x > 1

Evaluate the integrals in Exercises 31–78.

71. ∫(from √2/3 to 2/3)dy/(|y|√(9y²-1))

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

32. lim (x → 0) (3^x - 1) / (2^x - 1)

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

83. y = 3^(log₂ t)

Evaluate the integrals in Exercises 53–76.

61. ∫(from 0 to 2)dt/√(8+2t²)

Find the values in Exercises 9–12.

11. tan(arcsin(-1/2))

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

23. y=arcsin(√2t)

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

71. y = log₂(5θ)

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

16. y = (ln x)³

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

17. lim(x→∞)arcsec(x)

Evaluate the integrals in Exercises 33–54.

∫ 2t e^(-t²) dt

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

85. y = log₂(8t^(ln 2))

Solve the initial value problems in Exercises 55–58.

57. d²y/dx² = 2e^(−x), y(0) = 1, y′(0) = 0

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

73. y = log₄ x + log₄ x²

Evaluate the integrals in Exercises 39–56.

56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))

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)

Evaluate the integrals in Exercises 53–76.

63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))

Evaluate the integrals in Exercises 97–110.

109. ∫ (dx / (x log₁₀x))

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)

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→∞.

Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.

4. cosh x = 13/5, x>0