$\text{ If }{\int }_0^{\mathrm{\pi }} \frac{\sin \mathrm{x}(\sin \mathrm{x}+1){\mathrm{e}}^{\sin \mathrm{x}+\cos \mathrm{x}}}{{\mathrm{e}}^{\cos \mathrm{x}}+1}\mathrm{dx}=\mathrm{a}+\mathrm{b}{\int }_0^{\mathrm{\pi }} {\mathrm{e}}^{\sin \mathrm{x}}\mathrm{dx}\text{ where a and }\mathrm{b}\text{ are positive rational }$

$\text{ numbers, then the value of }100\left(a^2+b^2\right)\text{ is }$

$\begin{array}{l}\mathrm{I}={\int }_0^{\mathrm{\pi }} \frac{\sin \mathrm{x}(1+\sin \mathrm{x}){\mathrm{e}}^{\sin \mathrm{x}-\cos \mathrm{x}}}{{\mathrm{e}}^{-\cos \mathrm{x}}+1}\mathrm{dx}\\ \mathrm{I}={\int }_0^{\mathrm{\pi }} \frac{\sin \mathrm{x}(1+\sin \mathrm{x}){\mathrm{e}}^{\sin \mathrm{x}}}{{\mathrm{e}}^{\cos \mathrm{x}}+1}\mathrm{dx}-----\left(1\right)\end{array}$

$\begin{array}{l}\text{ Add }2\mathrm{I}=\frac{1}{2}{\int }_0^{\mathrm{\pi }} \frac{\sin \mathrm{x}(1+\sin \mathrm{x}){\mathrm{e}}^{\sin \mathrm{x}}\left({\mathrm{e}}^{\cos \mathrm{x}}+1\right)}{{\mathrm{e}}^{\cos \mathrm{x}}+1}\mathrm{dx}\\ \mathrm{I}=\frac{1}{2}{\int }_0^{\mathrm{\pi }} \sin \mathrm{x}(1+\sin \mathrm{x}){\mathrm{e}}^{\sin \mathrm{x}}\mathrm{dx}\\ \mathrm{I}=\frac{1}{2}{\int }_0^{\mathrm{\pi }} {\mathrm{e}}^{\sin \mathrm{x}}\left(\sin \mathrm{x}+1-{\cos }^2\mathrm{x}\right)\mathrm{dx}\\ \mathrm{I}=\frac{1}{2}{\int }_0^{\mathrm{\pi }} {\mathrm{e}}^{\sin \mathrm{x}}\mathrm{dx}+\frac{1}{2}{\int }_0^{\mathrm{\pi }} {\mathrm{e}}^{\sin \mathrm{x}}\left(\sin \mathrm{x}-{\cos }^2\mathrm{x}\right)\mathrm{dx}\\ \mathrm{I}=\frac{1}{2}{\int }_0^{\mathrm{\pi }} {\mathrm{e}}^{\sin \mathrm{x}}\mathrm{dx}+\frac{1}{2}{\left[{\mathrm{e}}^{\sin \mathrm{x}}(-\cos \mathrm{x})\right]}_0^{\mathrm{\pi }}\\ \mathrm{I}=\frac{1}{2}{\int }_0^{\mathrm{\pi }} {\mathrm{e}}^{\sin \mathrm{x}}\mathrm{dx}+\frac{1}{2}[1+2]\\ \mathrm{I}=1+\frac{1}{2}{\int }_0^{\mathrm{\pi }} {\mathrm{e}}^{\sin \mathrm{x}}\mathrm{dx}\\ 100\left(1+\frac{1}{4}\right)=125\end{array}$