$\frac{\mathrm{d}}{\mathrm{dx}}\left({\int }_{{\mathrm{x}}^2}^{{\mathrm{x}}^3} \frac{1}{\log \mathrm{t}}\mathrm{dt}\right)$ is equal to

Solution:

$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{dx}}\left({\int }_{{\mathrm{x}}^2}^{{\mathrm{x}}^3} \frac{1}{\log \mathrm{t}}\cdot \mathrm{dt}\right)=\frac{1}{\log {\mathrm{x}}^3}\cdot \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x}{)}^3-\frac{1}{\log {\mathrm{x}}^2}\cdot \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{x}}^2\right)\\ =\frac{3{\mathrm{x}}^2}{3\log \mathrm{x}}-\frac{2\mathrm{x}}{2\log \mathrm{x}}\\ \Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\left({\int }_{{\mathrm{x}}^2}^{{\mathrm{x}}^3} \frac{1}{\log \mathrm{t}}\mathrm{dt}\right)=\frac{1}{\log \mathrm{x}}\cdot \left({\mathrm{x}}^2-\mathrm{x}\right)\end{array}$