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

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) = x² − 2x, x ≤ 1

Solve the initial value problems in Exercises 87 and 88.

87. dy/dx = 1 + 1/x, y(1) = 3

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

59. y = √(t/(t+1))

73. Find the area between the curves y=ln(x) and y=ln(2x) from x=1 to x=5.

128. Derive the formula dy/dx = 1/(1+x²) for the derivative of y = arctan(x) by differentiating both sides of the equivalent equation tan(y)=x.

Solve the differential equation in Exercises 9–22.

12. (dy/dx) = 3x²e^(-y)

Evaluate the integrals in Exercises 39–56.

49. ∫3sec²t/(6 + 3tan(t)) dt

Evaluate the integrals in Exercises 53–76.

71. ∫(from -π/2 to π/2) 2cosθ dθ/(1+(sinθ)²)

Evaluate the integrals in Exercises 39–56.

39. ∫(from -3 to -2)dx/x

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)

In Exercises 73 and 74, repeat the steps above to solve for the functions y=f(x) and x=f^(-1)(y) defined implicitly by the given equations over the interval.

73. y^(1/3) - 1 = (x+2)³, -5 ≤ x ≤ 5, x_0 = -3/2

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 41–60.

45. ∫tanh(x/7)dx

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

27. y = (1 - θ)tanh⁻¹(θ)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

53. lim (x → 1⁺) x^(1/(1 - x))

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

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

Evaluate the integrals in Exercises 77–90.

81. ∫dy/(y²-2y+5)

Evaluate the integrals in Exercises 91–102.

93. ∫(arcsin x)²dx/√(1-x²)

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

67. y = 7^(sec θ) ln 7"

Evaluate the integrals in Exercises 53–76.

67. ∫dx/(2+(x-1)²)

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 77–90.

90. ∫dx/((x-2)√(x²-4x+3))

Evaluate the integrals in Exercises 33–54.

51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv

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

35. y=arccsc(e^t)

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x

Evaluate the integrals in Exercises 91–102.

99. ∫1/(√x (x+1)((arctan√x)²+9)) dx

Evaluate the integrals in Exercises 33–54.

∫(e^(3x) + 5e^(-x)) dx

Solve the differential equation in Exercises 9–22.

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