math - Laplace Transform of $-e^{-at}u(-t)$

I have found a problem in applying Laplace Transform to $-e^{-at}u(-t)$ I am doing these steps:

$$ = - \int_{-\infty}^{+\infty} e^{-at}u(-t) e^{-st}dt$$ $$ = - \int_{-\infty}^{0} e^{-at} e^{-st}dt$$ $$ = - \int_{-\infty}^{0} e^{-(a+s)t}dt$$ $$ = - [-\frac{1}{a+s} e^{-(a+s)t}]|_{-\infty}^{0}$$ $$ = - [-\frac{1}{a+s} (e^{-(a+s)0}-e^{-(a+s)-\infty})]$$

$$ = - [-\frac{1}{a+s} (1- \infty)]$$

$$ = \infty$$ Can anyone help me why it is showing like that.I check it on internet and all the books are showing the answer is $\frac{1}{s+a}$