Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.

f(x) = (1 − x)³

Evaluate the integrals in Exercises 87–96.

91. ∫₁^(√2) x·2^(x²) dx

Evaluate the integrals in Exercises 33–54.

∫(from ln3 to ln2) (e^x) dx

In Exercises 27–32, find dy/dx.

e^(2x)=sin(x+3y)

Evaluate the integrals in Exercises 39–56.

56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))

In Exercises 1–4, solve for t.

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

Evaluate the integrals in Exercises 97–110.

103. ∫₁⁴ (ln 2 · log₂x / x) dx

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

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

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(5-7x)

Evaluate the integrals in Exercises 111–114.

113. ∫₁^(1/x) (1 / t) dt, x > 0

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 39–56.

55. ∫dx/(2√x + 2x)

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

25. y=arcsec(2s+1)

126. Show that the sum arctan(x)+arctan(1/x) is constant.

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

29. y = (1 - t)coth⁻¹(√t)

Evaluate the integrals in Exercises 41–60.

51. ∫(from ln2 to ln4)coth(x)dx

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

57. lim (x → 0⁺) x^(-1/ln x)

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))

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(-5x)

Evaluate the integrals in Exercises 97–110.

99. ∫₀³ (√2 + 1)x^(√2) dx

Evaluate the integrals in Exercises 97–110.

109. ∫ (dx / (x log₁₀x))

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = ln(3te^(-t))

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

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

9. lim (t → -3) (t³ - 4t + 15) / (t² - t - 12)

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)

Evaluate the integrals in Exercises 53–76.

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

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

43. y=√(arcsin x)

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

33. y = csch⁻¹(1/2)^θ

Solve the differential equation in Exercises 9–22.

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