Mathematics - Definite Integration Question with Solution | TestHub

Let $\mathrm{y}={\lim }_{\mathrm{n}\to \infty }\frac{1}{{\left({\mathrm{n}}^2\right)}^{\mathrm{m}}}{\left[{\prod }_{\mathrm{r}=1}^{\mathrm{n}}{\left({\mathrm{n}}^2+{\mathrm{r}}^2\right)}^{\mathrm{m}}\right]}^{\frac{1}{\mathrm{n}}}$

$$\begin{array}{rlr}& \ell \mathrm{ny}=\underset{\mathrm{n}\to \infty }{\lim }\mathrm{m}\sum _{\mathrm{r}=1}^{\mathrm{n}}\frac{\ell \mathrm{n}\left(\frac{{\mathrm{n}}^2+{\mathrm{r}}^2}{{\mathrm{n}}^2}\right)}{\mathrm{n}}& \end{array}$$

$$=\underset{\mathrm{n}\to \infty }{\lim }\sum _{\mathrm{r}=1}^{\mathrm{n}}\frac{\log \left(\frac{{\mathrm{n}}^2+{\mathrm{r}}^2}{{\mathrm{n}}^2}\right)}{\mathrm{n}}={\int }_0^1\log \left(1+{\mathrm{x}}^2\right)\mathrm{dx}=\log 2+\frac{\pi }{2}-2(given)$$

So, $\ell \mathrm{ny}=\mathrm{m}\left(\log 2+\frac{\pi }{2}-2\right)$

$$\mathrm{y}={\mathrm{e}}^{\mathrm{m}\log 2}{\mathrm{e}}^{\mathrm{m}\left(\frac{\pi }{2}-2\right)}$$