Question Number 102121 by Rohit@Thakur last updated on 06/Jul/20

∫_(−∞) ^∞ ((sin(x+(π/2)))/(1+x^2 )) dx By real analysis

Commented byprakash jain last updated on 06/Jul/20

https://youtu.be/YWBdwYr6PGg Solution video.

Answered by Ar Brandon last updated on 06/Jul/20

Let f(x)=∫_(−∞) ^∞ ((sin(x+(π/2)))/(1+x^2 ))dx=∫_(−∞) ^(+∞) ((sin((π/2)−x))/(1+x^2 ))dx ⇒f(x)=∫_(−∞) ^∞ ((cosx)/(1+x^2 ))dx ⇒f(−x)=∫_(−∞) ^∞ ((cosx)/(1+x^2 ))dx ⇒f(x)=2∫_0 ^∞ ((cosx)/(1+x^2 ))dx Any idea to proceed ?

Answered by Dwaipayan Shikari last updated on 06/Jul/20

∫_(−∞) ^∞ ((sin(x+(π/2)))/(1+x^2 ))dx=∫_(−∞) ^∞ ((cosx)/(1+tan^2 θ))sec^2 θdθ { take x as tanθ ∫_(−(π/2)) ^(π/2) cos(tanθ)dθ ....continue

Answered by mathmax by abdo last updated on 06/Jul/20

sir rohit want real method...!

Commented byprakash jain last updated on 06/Jul/20

Yez. I had this vidro link saved so i shared it anyway.