Evaluate the integrals in Exercises 53–76.
67. ∫dx/(2+(x-1)²)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

53. lim (x → 1⁺) x^(1/(1 - x))

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) = 27x³

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

44. lim (x → 0⁺) (csc x - cot x + cos x)

80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

15. y = 2√t tanh(√t)

Evaluate the integrals in Exercises 41–60.

49. ∫(sech(√t)tanh(√t)dt)/√t