Evaluate the integrals in Exercises 77–90.

79. ∫(from -1 to 0)6dt/√(3-2t-t²)

Evaluate the integrals in Exercises 87–96.

91. ∫₁^(√2) x·2^(x²) dx

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

47. y=(arccot(x³))³

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.

73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)

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

24. lim (x → π/2) (ln(csc x)) / (x - (π/2))²

Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

25. y = sinh⁻¹(√x)

Evaluate the integrals in Exercises 33–54.

∫(from ln3 to ln2) (e^x) dx

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

71. y = log₂(5θ)

86. Use a derivative to show that g(x)=√(x² + ln x) is one-to-one.

Evaluate the integrals in Exercises 97–110.

99. ∫₀³ (√2 + 1)x^(√2) dx

In Exercises 1–4, solve for t.

e^(sqrt(t)) = x^2

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))

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

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

Evaluate the integrals in Exercises 33–54.

∫ (e^(√r) / √r) dr

Evaluate the integrals in Exercises 97–110.

105. ∫₀² (log₂(x + 2) / (x + 2)) dx

In Exercises 5 and 6, solve for t.

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

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

70. y = ∛(x(x+1)(x-2)/(x²+1)(2x+3))

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

57. lim (x → 0⁺) x^(-1/ln x)

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

77. y = log₃(((x + 1)/(x − 1))^(ln 3))

Find the values in Exercises 9–12.

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

For problems 49–52 use implicit differentiation to find dy/dx at the given point P.

51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)

Evaluate the integrals in Exercises 87–96.

87. ∫ 5ˣ dx

Solve the initial value problems in Exercises 115–120.

115. dy/dx = 1/√(1 - x²), y(0) = 0

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

19. lim(x→∞)arccsc(x)

Solve the initial value problems in Exercises 55–58.

55. dy/dt = e^t sin(e^t − 2), y(ln 2) = 0

Evaluate the integrals in Exercises 53–76.

73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))

84. Find lim(x→∞) (√(x² + 1) - √x).

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

14. lim (t → 0) sin 5t / 2t

Solve the differential equation in Exercises 9–22.

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