Evaluate the integrals in Exercises 77–90.

79. ∫(from -1 to 0)6dt/√(3-2t-t²)

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

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

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→∞.

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))

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)

Evaluate the integrals in Exercises 39–56.

49. ∫3sec²t/(6 + 3tan(t)) dt

Solve the differential equation in Exercises 9–22.

12. (dy/dx) = 3x²e^(-y)

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

Evaluate the integrals in Exercises 53–76.

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

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

49. 3arctan(x) + arcsin(y) = π/4; P(1, -1)

Evaluate the integrals in Exercises 33–54.

∫ (e^(√r) / √r) dr

Evaluate the integrals in Exercises 53–76.

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

Evaluate the integrals in Exercises 111–114.

111. ∫₁^(ln x) (1 / t) dt, x > 1

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

45. y=cos(x-arccos(x))

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

y = e^(5-7x)

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

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

29. y = (1 - t)coth⁻¹(√t)

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

23. y = ln(x)/(1+ln(x))

Evaluate the integrals in Exercises 33–54.

∫8e^(x+1) dx

43. Surrounding medium of unknown temperature A pan of warm water (46°C) was put in a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 min after that, it was 33°C. Use Newton’s Law of Cooling to estimate how cold the refrigerator was.

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

81. y = log₁₀(e^x)

"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 87–96.

87. ∫ 5ˣ dx

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 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

119. y = (sin x)ˣ

47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?

Evaluate the integrals in Exercises 39–56.

45. ∫(from 1 to 2)(2ln x)/x dx

Solve the differential equation in Exercises 9–22.

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