Question

The Laplace transform of e^-at sin ωt u(t) is:
1. \(\frac{\omega }{{{{\left( {s + a} \right)}^2} + {\omega ^2}}}\)
2. \(\frac{\omega }{{\left( {s + a} \right) + \omega }}\)
3. \(\frac{{s + a}}{{\left( {s + a} \right) + \omega }}\)
4. \(\frac{{s + a}}{{{{\left( {s + a} \right)}^2} + {\omega ^2}}}\)

Answer 1

Correct Answer - Option 1 : \(\frac{\omega }{{{{\left( {s + a} \right)}^2} + {\omega ^2}}}\)

Concept:

Bilateral Laplace transform:

\(L\left[ {x\left( t \right)} \right] = x\left( s \right) = \;\mathop \smallint \limits_{ - \infty }^\infty x\left( t \right){e^{ - st}}dt\)

Unilateral Laplace transform:

\(L\left[ {x\left( t \right)} \right] = x\left( s \right) = \;\mathop \smallint \limits_0^\infty x\left( t \right){e^{ - st}}dt\)

Some important Laplace transforms:

	f(t)	f(s)	ROC
1.	δ(t)	1	Entire s-plane
2.	e-at u(t)	\(\frac{1}{{s + a}}\)	s > - a
3.	e-at u(-t)	\(\frac{1}{{s + a}}\)	s < - a
4.	cos ω0 t u(t)	\(\frac{s}{{{s^2} + \omega _0^2}}\)	s > 0
5.	te-at u(t)	\(\frac{1}{{{{\left( {s + a} \right)}^2}}}\)	s > - a
6.	sin ω0t u(t)	\(\frac{{{\omega _0}}}{{{s^2} + \omega _0^2}}\)	s > 0
7.	u(t)	1/s	s > 0

 

Calculation:

\(\sin \omega t. u(t)\leftrightarrow \frac{\omega }{{{s^2} + {\omega ^2}}}\)

By applying frequency differentiation property,

\({e^{ - at}}\sin \omega t. u(t) \leftrightarrow \frac{\omega }{{{{\left( {s + a} \right)}^2} + {\omega ^2}}}\)