143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.

Evaluate the integrals in Exercises 111–114.

113. ∫₁^(1/x) (1 / t) dt,x > 0

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

15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.

Evaluate the integrals in Exercises 47–68.

∫₀^π/3 sec² θ dθ

Evaluate the integrals in Exercises 47–68.

∫⁰-π/3 sec x tan x dx

Given the definite integral $F\left(x\right)={\int }_3^x[t^8-\sin \left(t^4\right)]dt$ , find the derivative $F^{\prime }\left(x\right)$.