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Can this integeral be evaluated? $$\int_0^\infty\cos(ax)\sec(bx)\mathrm{d}x$$

Well, this problem came from the integral below: $$\int_0^\infty\cos(ax)\mathrm{sech}(bx)\mathrm{d}x$$ where $\mathrm{sech}(\cdot)$ is the hyperbolic secant function, i.e., $\mathrm{sech}(x)=\dfrac{2}{\mathrm{e}^x+\mathrm{e}^{-x}}$. I found the common solution is by Residual Theorem like this one. From my perspective, this may be resolved if there is a way to evaluate $\int_0^\infty\cos(ax)\sec(bx)\mathrm{d}x$ since $\mathrm{sech}(x)=\sec(\mathrm{i}x)$. At this point, the problem can be reduced to evaluate $\int_0^\infty\cos(ax)\sec(\mathrm{i}bx)\mathrm{d}x$.

So how can I evaluate $\int_0^\infty\cos(ax)\sec(bx)\mathrm{d}x$?

Gary
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Jasmine
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    You write the question as if you are assigning us homework. This doesn't go over big here. Better to tell us where this came up, what tools you have for doing this sort of question, where you got stuck, etc., etc. – Gerry Myerson Mar 10 '22 at 11:33
  • @GerryMyerson Sorry, I have updated the description! – Jasmine Mar 10 '22 at 12:10
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    If $a/b$ is rational or $b=0$, the integrand is periodic, so the integral diverges. What do you think happens when $a/b$ is irrational? – J.G. Mar 10 '22 at 12:23
  • Distantly related: https://math.stackexchange.com/questions/1759905/evaluate-int-0-infty-frac-sinh-bx-sinh-ax-dx and https://math.stackexchange.com/questions/1657473/prove-int-0-infty-fracxk-1-x-k-1xa-x-adx-frac-pia-c and https://math.stackexchange.com/questions/804680/show-int-0-infty-frac-cos-a-x-cos-b-x-sinh-beta-x-fracdxx-log-big and https://math.stackexchange.com/questions/334567/evaluate-the-integral-using-contour-integration-theorem-of-residues – Gerry Myerson Mar 10 '22 at 21:50
  • Were any of those links helpful, Jas? – Gerry Myerson Mar 12 '22 at 05:24
  • I’m voting to close this question because OP does not engage. – Gerry Myerson Mar 13 '22 at 11:46

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