Only Ratio Matters!!!

$Here$

$I_{1}= \int _{0}^{\frac{ \pi }{2}}\sqrt[]{1+\frac{1}{\sqrt[]{1+\cot ^{2011}x}}}dx$

$I_{2}= \int _{0}^{\frac{ \pi }{2}}\sqrt[]{1-\frac{1}{\sqrt[]{1+\cot ^{2011}x}}}dx$

$Let~\frac{1}{\sqrt[]{1+\cot ^{2011}x}}=\sin \theta ~ \text{for some} ~ \theta , since$

$\frac{1}{\sqrt[]{1+\cot ^{2011}x}}\text{<1 for x in } \left( 0,\frac{ \pi }{2} \right)$

$\text{It can be easily proved that }$

$\cos \theta =\frac{\sqrt[]{\cot ^{2011}x}}{\sqrt[]{1+\cot ^{2011}x}},using~\sin ^{2} \theta +\cos ^{2} \theta =1.$

$also,~I_{1}= \int _{0}^{\frac{ \pi }{2}}\sqrt[]{1+\frac{1}{\sqrt[]{1+\cot ^{2011} \left( \frac{ \pi }{2}-x \right) }}}dx$

$I_{1}= \int _{0}^{\frac{ \pi }{2}}\sqrt[]{1+\frac{\sqrt[]{\cot ^{2011}x}}{\sqrt[]{1+\cot ^{2011}x}}}dx~$

$Therefore,$

$2I_{1}= \int _{0}^{ \pi /2} \left( \sqrt[]{1+\frac{1}{\sqrt[]{1+\cot ^{2011}x}}}+\sqrt[]{1+\frac{\sqrt[]{\cot ^{2011}x}}{\sqrt[]{1+\cot ^{2011}x}}} \right) dx$

$\text{or, 2}I_{1}= \int _{0}^{ \pi /2} \left( \sqrt[]{1+sin \theta }+\sqrt[]{1+cos \theta } \right) dx \ldots .. \left( 1 \right)$

$\text{similarly, 2}I_{2}= \int _{0}^{ \pi /2} \left( \sqrt[]{1-sin \theta }+\sqrt[]{1-cos \theta } \right) dx \ldots .. \left( 2 \right)$

$\text{ Now, for all}~ \theta~\text{ in }~ \left( 0,\frac{ \pi }{2} \right)$

$\frac{\sqrt[]{1+sin \theta }+\sqrt[]{1+cos \theta }}{\sqrt[]{1-sin \theta }+\sqrt[]{1-cos} \theta }=\sqrt[]{2}+1$

$\text{and hence it follows that}$

$\frac{I_{1}}{I_{2}}=\sqrt[]{2}+1$

$Hence~ \left( \frac{I_{1}}{I_{2}} \right) ^{2}+ \left( \frac{I_{2}}{I_{1}} \right) ^{2}=3+2\sqrt[]{2}+3-2\sqrt[]{2}=6.$