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I am looking for alternate integral representation of

$$\sum_{k=1}^\infty\int_0^\infty \frac{\sin ((k+1)x)\sin (kx)}{x^2k^4}dx \tag{1}\label{1}$$

After trying this for some time it doesn't seem like it has a nice integral representation. I'm least interested in evaluating the integral, I'm wondering if $\eqref{1}$ can be written in a nice integral form.

Any help or hints? Thanks in advance.

BooleanCoder
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  • It can be. The reference will help :) https://math.stackexchange.com/questions/106570/how-do-i-show-that-int-infty-infty-frac-sin-x-sin-nxx2-dx-pi – Svyatoslav Apr 15 '21 at 05:32
  • The solution in the post is given for arbitrary $a>b>0$: $2\int_0^{\infty}\frac{\sin(ax)\sin(bx)}{x^2};dx=\pi b$ – Svyatoslav Apr 15 '21 at 05:40
  • I'm looking for an alternate representation not a simplified version of it – BooleanCoder Apr 15 '21 at 05:41

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