Mathematics - Definite Integration Question with Solution | TestHub

Let $\mathrm{y}={\left(\mathrm{ex}\right)}^{\mathrm{x}}$

$\mathcal{l}\mathrm{ny}=x\left[1+\mathcal{l}\mathrm{nx}\right]$

$\frac{1}{\mathrm{y}}\frac{\mathrm{dy}}{\mathrm{dx}}=\left(2+\mathcal{l}\mathrm{nx}\right)$

$\Rightarrow$ $\mathrm{dy}={\left(\mathrm{ex}\right)}^{\mathrm{x}}\left(2+\mathcal{l}\mathrm{nx}\right) \mathrm{dx}$

$\underset{1}{\overset{2}{\int }}{\mathrm{e}}^{\mathrm{x}}.{\mathrm{x}}^{\mathrm{x}}\left(2+{\log }_{\mathrm{e}}\mathrm{x}\right)\mathrm{dx}$

$={\left(\mathrm{y}\right)}_1^2$

$={\left({\left(\mathrm{ex}\right)}^{\mathrm{x}}\right)}_1^2$

$=4{\mathrm{e}}^2-\mathrm{e}$