If$\int \frac{(\sqrt{\mathrm{x}}{)}^5\mathrm{dx}}{(\sqrt{\mathrm{x}}{)}^7+{\mathrm{x}}^6}=\mathrm{\lambda log}\left(\frac{{\mathrm{x}}^{\mathrm{a}}}{{\mathrm{x}}^{\mathrm{a}}+1}\right)+\mathrm{C}$ then $\mathrm{a}+\mathrm{\lambda }$

$\mathrm{I}=\int \frac{\mathrm{dx}}{(\sqrt{\mathrm{x}}{)}^2+(\sqrt{\mathrm{x}}{)}^7}=\int \frac{\mathrm{dx}}{(\sqrt{\mathrm{x}}{)}^7\left(\frac{1}{(\sqrt{\mathrm{x}}{)}^5}+1\right)}$

Put $1+\frac{1}{(\sqrt{\mathrm{x}}{)}^5}=\mathrm{t}\Rightarrow -\frac{5}{2}\cdot \frac{1}{(\sqrt{\mathrm{x}}{)}^7}\mathrm{dx}=\mathrm{dt}$

$\begin{array}{l}\mathrm{I}=-\frac{2}{5}\int \frac{\mathrm{dt}}{\mathrm{t}}=-\frac{2}{5}\log \mathrm{t}+\mathrm{C}\\ =-\frac{2}{5}\log \left[1+\frac{1}{(\sqrt{\mathrm{x}}{)}^5}\right]+\mathrm{C}=\frac{-2}{5}\log \left(\frac{{\mathrm{x}}^{5/2}+1}{{\mathrm{x}}^{5/2}}\right)+\mathrm{C}\end{array}$

$\Rightarrow \mathrm{\lambda }=\frac{-2}{5}\text{ and }\mathrm{a}=\frac{5}{2}$

$\therefore \mathrm{a}+\mathrm{\lambda }=\frac{21}{10}>2$