The derivative of $F\left(x\right)=\underset{x^2}{\overset{x^3}{\int }}\frac{1}{\log t}\cdot dt(x>0)=$

$F\left(x\right)=\underset{x^2}{\overset{x^3}{\int }}\frac{1}{\log t}\cdot dt(x>0)$

Newton Leibnitz formula :

$\frac{d}{dx}(\underset{\varphi \left(x\right)}{\overset{\Psi \left(x\right)}{\int }}f\left(t\right)\cdot dt=f\left(\Psi \left(x\right)\right)\frac{d}{dx}\left(\Psi \left(x\right)\right)-f\left(\varphi \left(x\right)\right)\cdot \frac{d}{dx}\left(\varphi \left(x\right)\right))$

$=\frac{1}{\log x^3}\cdot 3x^2-\frac{1}{\log x^2}\cdot 2x$

$=\frac{3x^2}{3\log x}-\frac{2x}{2\log x}$

$=\frac{x^2}{\log x}-\frac{x}{\log x}=\frac{x(x-1)}{\log x}=x(x-1)\cdot {\left(\log x\right)}^{-1}$.