Question

Calculate the derivative

using Part 2 of the Fundamental Theorem of Calculus.

Answer 1

Explanation:

`(d)/(dx)\int_(x^3)^{e^(2x)}ln(t)dt=(d)/(dx)\left(F(e^(2x))-F(x^3)\right)=F'(e^(2x))(e^(2x))'-F'(x^3)(x^3)'=\\=2e^(2x)F'(e^(2x))-3x^2F'(x^3)=2e^(2x)\ln(e^(2x))-3x^2\ln(x^3)=4xe^(2x)-9x^2\ln\abs x`