Evaluate the integrals in Exercises 111–114.
113. ∫₁^(1/x) (1 / t) dt,x > 0

Find dy/dx if y = ∫(From cos x to 0) 1/(1 - t²) dt.

Explain the main steps in your calculation.

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

37. ∫(from x²/2 to x²)ln(√t)dt

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

y = ∫(from 0 to lnx) sin(e^t) dt

Evaluate the integrals in Exercises 111–114.

111. ∫₁^(ln x) (1 / t) dt,x > 1

For Exercises 127 and 128 find a function f satisfying each equation.

128. f(x) = e² + ∫₁ˣ f(t) dt

For Exercises 127 and 128 find a function f satisfying each equation.

127. ∫₂ˣ √(f(t)) dt = x ln x

Evaluate the integrals in Exercises 111–114.

112. ∫₁^(eˣ) (1 / t) dt

143.

a. Show that ∫ ln(x) dx = x ln(x) − x + C.