I am working on the convolution below, however I have gotten stuck. I am not sure how to think about changing the bounds of the integrals to give me an answer. Here is the problem, where $u(t)$ is the Heaviside step function:
$$x(t) = u(t)-2u(t-2)+u(t-5) \\ h(t) = e^{2t}u(1-t)$$
and here is what I have done so far: \begin{align} x(t)*h(t) &= \int_{-\infty}^{\infty}(u(t)-2u(t-2)+u(t-5))h(t-\tau)d\tau \\ &= \int_0^2h(t-\tau)d\tau-\int_2^5h(t-\tau)d\tau \\ &= \int_0^2e^{2(t-\tau)}u(1-t+\tau)d\tau-\int_2^5e^{2(t-\tau)}u(1-t+\tau)d\tau \\ &= \space ??? \end{align}
Any advice is appreciated. I know that the solution has something to do with the fact that for $u(1-t-\tau), \space 1-t-\tau \ge 0$. I am not really sure how to change the bounds of these integrals to reflect this, however.
Thanks for any help that you can provide. Let me know if you need anything else.