Compute $\int_{-\infty}^{\infty} t \tan^{-1}(t) \exp (-t^2)\,dt$

Question

I would like to obtain $$ \int_{-\infty}^{\infty} t \tan^{-1}(t) \exp (-t^2)\,dt $$ My idea is to use Fourier transform and go with generalized Parseval. I choose $x_{1}(t)=\tan^{-1}(t)$ and $x_{2}^{*}(t)=t \exp(-t^2)$. I don't know how to proceed after. My problem with my idea is that I don't know the Fourier transform of the reciprocal function of tan(t). Any suggestion??

Or maybe another choice of functions for the Generalized Parseval theorem?

After integration by parts, you arrive at this problem. – metamorphy Jan 29 '21 at 13:13 — metamorphy, Jan 29 '21 at 13:13

score 2 · Accepted Answer · answered Jan 29 '21 at 13:15

2

I would do integration by parts first and then employ Fourier transform techniques.

$$\int \arctan(x)xe^{-x^2}\,dx = -\frac{1}{2}\arctan(x)e^{-x^2} + \frac{1}{2}\int \frac{1}{1+x^2}e^{-x^2}\,dx.$$

Evaluating at $\pm\infty$, the first term on the left hand side will go to $0$ since $\arctan$ is bounded and $e^{-x^2}$ converges to $0$ at infinity. The latter term is very amenable to Fourier transform techniques via Parseval. Can you take it from here?

answered Jan 29 '21 at 13:15

Cameron Williams

29,432

Thank you for the feedback. Right after this step, I suggest we go with generalized Parseval and some small transformations can lead us to the result. Am I correct? – why_me Jan 29 '21 at 13:49
Yep! That will do it. – Cameron Williams Jan 29 '21 at 13:59

Compute $\int_{-\infty}^{\infty} t \tan^{-1}(t) \exp (-t^2)\,dt$

1 Answers1