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Using contour integration, find $$\int_{0}^{+\infty}\frac{x^2\cos x}{\cosh x} \,{\rm d}x$$

How to calculate it? I never worked with integrals of this type.

I don't know how to rewrite the function(my photo)

Sumanta
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rpr
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    This is obviously an exercise. What have you tried to do? – KBS May 21 '22 at 19:22
  • @KBS So I found contour but don't know how to transform function(New link) – rpr May 22 '22 at 05:31
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    Hint: use $\displaystyle J(a)=\int_{0}^{+\infty}\frac{\cos ax}{\cosh x} ,dx=\Re,\int_{0}^{+\infty}\frac{e^{i ax}}{\cosh x} ,dx$, then your integral $I=-\frac{d^2}{da^2}J(a)\Big|_{a=1}$. For $J(a)$ use your rectangular contour: at $\displaystyle x\to x+\pi i\quad \frac{e^{i ax}}{\cosh x}\to -,e^{-\pi a}\frac{e^{i ax}}{\cosh x}$ – Svyatoslav May 22 '22 at 06:22
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    @Svyatoslav I am going to try it, thank you! (Are you from NSU :) ?) – rpr May 22 '22 at 07:59
  • You mean the Novosibirsk State University (NGU) in Akademgorodok? Yes; I was graduated in 1988 as a physicist :) – Svyatoslav May 22 '22 at 08:27
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    @Svyatoslav it was unexpected to see your hint, I'm currently studying at NSU :) – rpr May 22 '22 at 08:32
  • I wish you good luck! If you need some reasonable assistance, and I'm able help - I would be glad; please ask :) I wish you a nice day! – Svyatoslav May 22 '22 at 09:04
  • See https://math.stackexchange.com/questions/2544300/evaluating-int-0-infty-frac-cos-ax-coshxdx?rq=1 – xpaul May 23 '22 at 11:53
  • @xpaul I already solved it :) – rpr May 24 '22 at 07:06

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