Tuesday, January 7, 2020

Evaluate the integral: intinfty0fractan1(tx)xleft(1+x2right)mathrmdx



I've been trying to evaluate the integral for a while now, and I've been unable to find it anywhere...
I tried substituting tan1(tx) as u but got nowhere... I have done dozens of other substitutions...

I've been told to use the properties of gaussian integrals and the gamma function, but can't seem to figure out a way to bring in e...
A few hints that point in the right direction to think will really help!



It would be really really helpful if you could solve this using only the properties of gaussian and gamma functions!


Answer



Let:
I(t)=0arctantxx(1+x2)dx.
Then

I(t)=01(1+t2x2)(1+x2)dx=1t210[t21+t2x211+x2]dx=1t21[tarctan(tx)arctanx]0=π2|t|1t21=π21|t|+1.
Finally integrating the last expression over t one obtains:
I(t)=I(0)+π2sgn(t)log(|t|+1)=sgn(t)π2log(|t|+1).


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...