Help in Laplace Transform, not homework

I am trying to do laplace transform of [itex]y(t)=e^{-at}\cos (bt)\;[/itex]. The answer should be:

[tex]\frac {s+a}{(s+a)^2+b^2}[/tex]

But here is my work, I can't get rid of the -ve sign. I must be too blind to see the obvious, please help:

[tex]\overline{y(s)}=\int^∞_0 e^{-(s+a)t}\cos (bt) dt\;=\; \frac 1 b [e^{-(s+a)t}\sin(bt)]^∞_0\;+\frac {s+a}{b}\int^∞_0\;e^{-(s+a)t}\sin (bt) dt[/tex]

[tex][e^{-(s+a)t}\sin(bt)]^∞_0 =0 \;\Rightarrow\; \overline{y(s)}= \frac {s+a}{b}\int^∞_0\;e^{-(s+a)t}\sin (bt) dt\;=\;-\frac {s+a}{b^2} [e^{-(s+a)t}\cos(bt)]^∞_0\;-\frac {(s+a)^2}{b^2}\int^∞_0\;e^{-(s+a)t}\cos (bt) dt[/tex]

[tex][e^{-(s+a)t}\cos(bt)]^∞_0 =1 \;\Rightarrow\; \overline{y(s)}= -\frac {s+a}{b^2}-\frac {(s+a)^2}{b^2}\int^∞_0\;e^{-(s+a)t}\cos (bt) dt\;=\;-\frac {s+a}{b^2}-\frac {(s+a)^2}{b^2}\overline{y(s)}[/tex]

[tex]\Rightarrow\;\left(1+\frac {(s+a)^2}{b^2}\right)\overline{y(s)}\;=\;-\frac {s+a}{b^2}\;\Rightarrow\; \overline{y(s)}\;=\; -\frac {s+a}{(s+a)^2+b^2}[/tex]

You see the -ve sign? Please tell me what I did wrong.

Thanks

 

Thanks for the reply. But on the second integral it is
[tex]-\int wdu\;\hbox { where }\; u= e^{-(s+a)t} \Rightarrow du=-(s+a)e^{-(s+a)t} dt\;,\; dw=\sin(bt)dt\;\Rightarrow \;w=-\frac 1 b \cos (bt)[/tex]

[tex]\Rightarrow\; -\int wdu\;=\; -\frac {-(s+a)}{-b}\int^∞_0 e^{-(s+a)t} \cos(bt)dt[/tex]

You see the -ve sign holds?

I just did it the first way come to mind to prove the formula, not working as an exercise. Now I got stuck in this!!! I gone through so many times and just can't see what I did wrong!!!

 

I don't see it. I look at this for a long time. I modified the last post to reflect the -ve signs and still the same. Please point it out. I know once a while, I get into this and I just cannot see it. This is supposed to be an easy proof of the formula.

 

Yes, I actually did the problem using complex Euler formula already. But that does not answer the question why I can't get the right sign. I check it so many times. Whether the way I do this is the most efficient way is besides the point. The point is why don't I get the right answer using simple integrate by parts. No matter what, I should get the correct answer. Right?

 

Anyone please?