Use the second fundamental theorem of calculus to compute?

Question

d/x, integral sign x^2 on top x^3 on bottom, dx/lnx when x=1/2.

Answer 1

Letting F(x) be an antiderivative of 1/ln x (it is a continuous function everywhere it is defined, so it has one -- indeed, the second FTC states that [a, x]∫1/ln x dx is such an antiderivative), we have:

d/dx ([x³, x²]∫1/ln x dx)
d/dx (F(x²) - F(x³))
2x F'(x²) - 3x² F'(x³)
2x/ln (x²) - 3x²/ln (x³)
2x/(2 ln x) - 3x²/(3 ln x)
(x - x²)/ln x

And we are done.