Evaluate:
$$ \int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx $$

Question

Evaluate:
$$ \int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx $$

anonymous · Answer

integral calculus?

experimentx · Answer

yep ... very nice problem!!

anonymous · Answer

haha .. it's hard . hmm, wat grade r u in?

experimentx · Answer

university!!

anonymous · Answer

sorry .. wat university?

experimentx · Answer

after school you go to university :D

anonymous · Answer

haha . FUNNY!

experimentx · Answer

hartnn · Answer

@experimentX does the solution involve taking Laplace Transform and in the end putting s=0 ?

experimentx · Answer

yeah i found the solution somewhat like that ...

experimentx · Answer

the other is to make use of Jordan' Lemma

experimentx · Answer

hartnn · Answer

even taking laplace of that function is difficult....

experimentx · Answer

currently I'm doing complex integration ... probably my last problem. moving to another chapter after this.

anonymous · Answer

$$I(a)=\int_0^\infty\frac{1-\cos ax}{x^2(x^2+1)}\,dx$$$$I''(a)=\int_0^\infty\frac{\cos ax}{1+x^2}\,dx$$

anonymous · Answer

by jordan$$I''(a)=\int_0^\infty\frac{\cos ax}{1+x^2}\,dx=\frac{\pi}{2e^a}$$?

anonymous · Answer

santosh?

experimentx · Answer

the latter part seems simple .. 
since the residue is only ... +i

anonymous · Answer

yeah ... is that right?im not sure

experimentx · Answer

let me evaluate it .. since the function is even ... |dw:1346060735592:dw|