How to prove that #int_0^oo##(e^(alphax)sinx)/x dx=cot^1alpha# given that #int_0^oo sinx/x dx = pi/2#?
1 Answer
May 19, 2018
See below
Explanation:
Liebnitz diff under the integral sign:
That is very doable on its own, but is also the Laplace transform of
#(dI)/(d alpha) = mathbb L_alpha ( sin x) =  1/(1 + alpha^2)#

We have an IV:
And from a trig identity for