by considering fourier transforms of
My advice, is obviously to calculate the fourier transform of these functions, and then use the convolution formula, but in this case use the fact that exp (i b k)=cos(bk)+isin(bk) to replace this with sin(bk) in the integrand.
if f(k),g(k) are the fourier transforms of f(x),g(x), and h(z) is the convolution of f and g then, then h(x)=F^-1(f(k)g(k)) where F^-1 is the inverse fourier transform.
The integral represents the value at x of (a multiple of) the inverse Fourier transform of the product of two functions, namely and . But the sinc function is the Fourier transform of a rectangular function, and the function is the Fourier transform of . So the inverse Fourier transform of their product will be the convolution of the rectangular function and that exponential function.
If and then .
That looks promising, because it gets the right answer apart from a constant multiple. However, I have cheated by using the function instead of . That doesn't matter provided that x is positive. This means that, in the integral, we want x–t to be positive whenever ; and that in turn means that I am assuming that . I claim that this is a reasonable assumption. Indeed, if it does not hold then the function will not have a Fourier transform (in the conventional sense: as the Captain points out, it will have a distributional Fourier transform of some sort), and so this whole method of evaluating the integral will become suspect.