How should I find a value of these integrals: $$ A:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{2-\nu}x^{\nu}}dx \quad\text{and}\quad B:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{1-\nu}x^{\nu}}dx, $$ where $\nu\in(0,1)$.
Because I asked about a problem as having resembled this before (here), I have tried to solve by the same argument, i.e., using the Laplace transform but I haven't found the formula for the Laplace transform. I also have attacked by the method via the complex integral (which appears in the Fourier transform) but it didn't work well. I guess that Bessel functions appear as the solution. However, I don't know how a solution (with Bessel functions) is provided if I do it.
I'm glad if you give a solution as possible in detail.
Thank you.