What is laplacetransform e^{-2t}sin(3pit) ?

The answer to laplacetransform e^{-2t}sin(3pit) is (3pi)/((s+2)^2+9pi^2)

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laplacetransform e^{-2t}sin(3pit)

laplace transform

e^{- 2 t} sin (3 π t)

\frac{3 π}{( s + 2 ) ^{2} + 9 π ^{2}}

Solution steps

L {e^{- 2 t} sin (3 π t)}

Apply transform rule:

L {f (t)} = F (s)

then

L {e^{a t} f (t)} = F (s - a)

For

e^{- 2 t} sin (3 π t) : f (t) = sin (3 π t), a = - 2

= L {sin (3 π t)} (s + 2)

L {sin (3 π t)} : \frac{3 π}{s ^{2} + 9 π ^{2}}

= \frac{3 π}{( s + 2 ) ^{2} + 9 π ^{2}}

\frac{d y}{d x} + 2 y = 50

\frac{d}{d t} (6 sin^{3} (t))

\frac{d ^{2}}{d x ^{2}} (\frac{5}{6 t ^{4}})

\frac{d y}{d x} = \frac{( x + y ) ^{2}}{2 x ^{2}}

\frac{d}{d x} (x^{2} - 4 x y + y^{2})

What is laplacetransform e^{-2t}sin(3pit) ?
The answer to laplacetransform e^{-2t}sin(3pit) is (3pi)/((s+2)^2+9pi^2)