Laplace transform of integral with Heaviside function

Question

I want to find the Laplace transform of
$$f(t)=\int_{0}^{t}\sin(r)u(r-\beta)(t-r)^{\alpha} dr$$ for $\alpha,\, \beta > 0$ where $u$ is the unit step function. I don't understand what my approach should be here – I know that $$\mathcal{L}\{u(t-c)f(t-c)\}=e^{-sc}\mathcal{L}\{f(t)\},$$ but I don't see how to convert the expression with the integral into the familiar form

Answer 1

Hint: $f(t) = \sin(t)u(t-\beta)\star t^\alpha$. Do you see where to go from here?

Edit: As a sanity check, your final answer should be:

$$ e^{-\beta s}\frac{\Gamma(\alpha+1)}{s^{\alpha+1}}\frac{\cos(\beta)+s\sin(\beta)}{s^2+1}$$