use integration by parts for I2
then a slight manipulation in the second term .
Quite easy after that.
\dpi{120} \hspace{-16}$If $\mathbf{I_{1}=\int_{0}^{\infty}(1+x^2)^{-2012}dx}$ and $\mathbf{I_{2}=\int_{0}^{\infty}(1+x^2)^{-2011}dx}$\\\\\\ Then find value of $\mathbf{\frac{I_{1}}{I_{2}}=}$
Ans=(4021)/(4022)
use integration by parts for I2
then a slight manipulation in the second term .
Quite easy after that.
Very similar to what asked in one of previous IIT's if I remember correctly.