Laplace transform $e^{at}$

Question

Ok the book says it is $\dfrac{1}{s-a}$ However when I evaluate $\displaystyle\int_0^{\infty}e^{-st}\cdot e^{at}=\displaystyle\int_0^{\infty}e^{-(s-a)t}$ so that the derivative is $\dfrac{-1}{s-a}=\dfrac{1}{a-s}$. what is wrong?

Answer 1

You can solve it also applying the shift theorem:

$$ \scr{L} \{e^{at}f(t)\}=F(s-a)$$ with $f(t)=1$ that becomes $F(s)=\frac{1}{s}$ .

So:
$$F(s-a)=\frac{1}{s-a}$$