Laplace Tranform ?!?!

**gabbee** · June 26th 2009, 06:00 PM

The question is laplace transform of 4sin(at+b) where a and b are constants. The answer is is ((a)cos(b)+(s)sin(b))*4/(s^2+a^2)-
i think table of integrals is used but how do they get to the answer.....

Help would be greatly appreciated

GAB

**Random Variable** · June 26th 2009, 07:02 PM

$\mathcal{L} [4 \sin(at+b)] = 4 \int_{0}^{\infty} e^{-st}\sin(at+b) \ dt$

$= 4 \int_{0}^{\infty} e^{-st} \Big(\sin(b) \cos(at) + \cos(b) \sin(at)\Big) \ dt$

$= 4 \sin(b) \int_{0}^{\infty} e^{-st} \cos(at) \ dt + 4 \cos(b) \int_{0}^{\infty} e^{-st} \sin(at) \ dt$

$4 \sin(b) \frac {s}{s^{2}+a^{2}} + 4 \cos(b) \frac {a}{s^{2}+a^{2}}$

$= \frac {4 \Big(a \cos(b) + s \sin(b) \Big) } {s^{2}+a^{2}}$

**gabbee** · June 26th 2009, 07:38 PM

Thanx tonnes fot that, it really really helped!!

xxx

**CaptainBlack** · June 26th 2009, 11:14 PM

Originally Posted by gabbee

The question is laplace transform of 4sin(at+b) where a and b are constants. The answer is is ((a)cos(b)+(s)sin(b))*4/(s^2+a^2)-
i think table of integrals is used but how do they get to the answer.....

Help would be greatly appreciated

GAB

The Laplace transform:

$[\mathcal{L}f](s)=\int_{-\infty}^{\infty} f(x) e^{-st}\ dt$

In this case:

$\mathcal{L}\left\{4 \sin(at+b)\right\}=4 \int_{-\infty}^{\infty} \sin(at+b) e^{-st}\ dt=$ $<br /> =4\ \text{Im} \left[ \int_{-\infty}^{\infty} e^{-st}e^{i(at+b)} \ dt \right] =$ $<br /> 4\ \text{Im} \left[e^{ib} \int_{-\infty}^{\infty} e^{(-s+ia)t} \ dt\right]$

and the rest is routine.

CB

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