I have the input signal $x(t)$ 
And impulse response $h(t)=20 e^{-1000t} u(t)$ in which u(t) is the unit step function.
When I try a convolution, I thought the solutions would be something like:
$ \begin{array}{ll} \int\limits_{0}^{t} 2\cdot 20e^{-1000(t-\tau)} = \frac{1}{25}(1-e^{-1000t}) & \mbox{if } 0 \le t < 2 \\ \int\limits_{0}^{2} 2\cdot 20e^{-1000(t-\tau)} + \int\limits_{2}^{t} -1\cdot 20e^{-1000(t-\tau)} = \frac{3}{50}e^{2000-1000t}-\frac{e^{-1000t}}{25}-\frac{1}{50} & \mbox{if } 2 \le t < 3 \\ \int\limits_{0}^{2} 2\cdot 20e^{-1000(t-\tau)} + \int\limits_{2}^{3} -1\cdot 20e^{-1000(t-\tau)} = \frac{3}{50}e^{2000-1000t}-\frac{1}{50}e^{3000-1000t}-\frac{e^{-1000t}}{25} & \mbox{if } t \geq 3 \\ \end{array} $
But this did not look like the answer in the solution manual, so did I do something wrong here?
Answer in solution manual: 
Alternative solution in solution manual: 
Your answers look equivalent to the solution manual up to a little algebra and/or rearrangements, since $\frac{1}{25} = \frac{2}{50}$ and $-1000(t-2) = 2000 - 1000t$.