Convolution calculation

Question

I'm trying to calculate the convolution between $x(t)=e^{-t}u(t)$ and $y(t)=e^{-(t-2)}u(t-2)$.

$\int_{-\infty}^{\infty} e^{-τ}u(τ)e^{t+τ-2}u(t+τ-2)dτ=\int_{0}^{t-2} e^{t-2}dτ=e^{t-2}\int_{0}^{t-2} dτ= e^{t-2} \left[τ\right]_0^{t-2}=e^{t-2}(t-2), t>2$

I tried to solve it using Laplace transformation and I'm getting as a result $e^{2-t}(t-2)$. I cannot figure out where I am mistaken (maybe I am mistaken in both solutions).

Answer 1

I've manually computed this and checked it using Mathematica. My thanks go to @bananapeel22 for pointing out the mistake. \begin{align} \quad&x(t)*y(t)\\ &=e^{-t}u(t)*e^{-(t-2)}u(t-2)\\ &=\int_{-\infty}^{\infty} e^{-τ}u(τ)e^{τ-t+2}u(t-τ-2)dτ\\ &=\int_{0}^{t-2} e^{2-t}dτ\\ &=(t-2)e^{2-t}u(t-2)\\ \end{align}