For Exercises 127 and 128 find a function f satisfying each equation.
128. f(x) = e² + ∫₁ˣ f(t) dt

Evaluate the integrals in Exercises 33–54.

49. ∫ e^(sec πt) sec πt tan πt dt

130. Use the identity arccot(u)=π/2 - arctan(u) to derive the formula for the derivative of arccot(u) in Table 7.4 from the formula for the derivative of arctan(u).

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

31. y = cos⁻¹(x) - x sech⁻¹(x)

77. The region in the first quadrant bounded by the coordinate axes, the line y=3, and the curve x=2/√(y+1) is revolved about the y-axis to generate a solid. Find the volume of the solid.

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.

1. sinh x = -3/4

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

47. lim (t → ∞) (e^t + t²) / (e^t - t)