$\mathcal{L}\left\{t \int_0^t \sin \tau d \tau \right\}$

Question

Can someone give me a hint on how to solve this? I know I can do $\mathcal{L}\left\{t\cdot 1 \ast \sin t\right\}$

I'm not allowed to evaluate any integrals before transforming in this section.

Answer 1

We shall use the following fact: If $(f, F)$ is a Laplace transform pair, then $$\mathcal{L} \left( tf(t) \right) = -F'(s) $$

(use entry 30 here). To that end, we let $f(t) = \int_0 ^\tau \sin \tau \ d\tau = (1 \star \sin(t))$ where $\star$ is the convolution operator. Now as the transform of a convolution is the product of its parts, we have $$\mathcal{L} \left(t (1 \star \sin (t) \right) =-\frac{d}{ds} \left( \frac{1}{s(s^2 + 1)} \right) $$

and from here it's just an application of the chain rule.