I think this question would be with complicated methods in singularity or other kind of methods. I need to solve the integral $$\int_0^\infty\frac{\cos(ax)-\cos(bx)}{x}dx$$ Thanks a lot.
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2Please show how have you tried – zed111 Mar 17 '15 at 07:44
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using Laplace transforms: $\log(|b/a|)$ – Math-fun Mar 17 '15 at 18:30
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A related question. – Lucian Mar 18 '15 at 00:51
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Hint: Either use the formula for $\cos A\pm\cos B$, which will ultimately result in an expression
similar to that of Dirichlet's integral, or employ a trick similar to Frullani's integral, namely by
noticing that $~\dfrac{\cos ax-\cos bx}x~=~\displaystyle\int_a^b\sin(tx)~dt,~$ and then switch the order of integration
via Fubini's theorem.
Lucian
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1@RonGordon: Just because it doesn't converge, doesn't mean that it cannot be evaluated... ;-$)$ – Lucian Mar 17 '15 at 13:55
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Yes, but it does mean you need to reconsider quoting Fubini's theorem. – Ron Gordon Mar 17 '15 at 14:06
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1@RonGordon: Or we could simply use it to justify the switch for a non-imaginary exponential integrand, and then mischievously and underhandedly generalize the real result for complex values of the exponent, via the Hand-Waving Theorem. – Lucian Mar 17 '15 at 14:15