TITLE: Solving A Definite Integral:The Feynman way
QUESTION [5 upvotes]: $ \displaystyle \int_0^\infty \dfrac{\cos xdx}{1+x^2} $                           
How to calculate the above using differentiation under the integral sign??

REPLY [2 votes]: Another way is to note that, if we put $$I\left(a\right)=\int_{0}^{\infty}\frac{\sin\left(ax\right)}{x\left(1+x^{2}\right)}dx,\,a>0$$ we have $$I'\left(a\right)=\int_{0}^{\infty}\frac{\cos\left(ax\right)}{1+x^{2}}dx
 $$ and now follow this answer to find $$I\left(a\right)=\frac{\pi}{2}\left(1-e^{-a}\right)$$ and so $$\lim_{a\rightarrow1}I'\left(a\right)=\frac{\pi }{2e}.$$