Evaluate the integrals in Exercises 33–54.
53. ∫ (e^r / (1 + e^r)) dr

Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).

In Exercises 27–32, find dy/dx.

3+siny = y-x^3

37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

63. lim (x → ∞) ((x + 2)/(x - 1))^x

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.

124. x^(sin y) = ln y

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

33. y=ln(arctan(x))