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)

Evaluate the integrals in Exercises 53–76.

61. ∫(from 0 to 2)dt/√(8+2t²)

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 111–114.

111. ∫₁^(ln x) (1 / t) dt, x > 1

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

32. lim (x → 0) (3^x - 1) / (2^x - 1)

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))

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))

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

73. y = log₄ x + log₄ x²

Solve the differential equation in Exercises 9–22.

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

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

85. y = log₂(8t^(ln 2))

Evaluate the integrals in Exercises 33–54.

∫(from ln4 to ln9)e^(x/2)dx

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

83. y = 3^(log₂ t)

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

71. y = log₂(5θ)

In Exercises 5 and 6, solve for t.

6. ln(t-2) = ln8 - ln(t)

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

23. y=arcsin(√2t)

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) = 1 - x/2, x ≤ 0

  x/(x + 2), x > 0

Evaluate the integrals in Exercises 97–110.

109. ∫ (dx / (x log₁₀x))

Evaluate the integrals in Exercises 33–54.

∫ 2t e^(-t²) dt

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)

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

Evaluate the integrals in Exercises 53–76.

63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))

Solve the initial value problems in Exercises 55–58.

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

Evaluate the integrals in Exercises 31–78.

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

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→∞.

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

17. lim(x→∞)arcsec(x)

Find the values in Exercises 9–12.

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