What is laplacetransform sin^3(t) ?

The answer to laplacetransform sin^3(t) is 6/((s^2+9)(s^2+1))

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laplacetransform sin^3(t)

laplace transform

sin^{3} (t)

\frac{6}{( s ^{2} + 9 ) ( s ^{2} + 1 )}

Solution steps

L {sin^{3} (t)}

Rewrite using trig identities

L {- \frac{1}{4} sin (3 t) + \frac{3}{4} sin (t)}

Use the linearity property of Laplace Transform:

For functions

f (t), g (t)

and constants

a, b : L {a \cdot f (t) + b \cdot g (t)} = a \cdot L {f (t)} + b \cdot L {g (t)}

= - \frac{1}{4} L {sin (3 t)} + \frac{3}{4} L {sin (t)}

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

L {sin (t)} : \frac{1}{s ^{2} + 1}

= - \frac{1}{4} \cdot \frac{3}{s ^{2} + 9} + \frac{3}{4} \cdot \frac{1}{s ^{2} + 1}

Simplify

- \frac{1}{4} \frac{3}{s ^{2} + 9} + \frac{3}{4} \frac{1}{s ^{2} + 1} : \frac{6}{( s ^{2} + 9 ) ( s ^{2} + 1 )}

= \frac{6}{( s ^{2} + 9 ) ( s ^{2} + 1 )}

\int x 5^{- x^{2}} d x

sec (x)

\frac{1}{( s - 2 ) ( s + 1 )}

(6, - 2), (12, 1)

x \to 2 lim (x^{2} + 3 x)

What is laplacetransform sin^3(t) ?
The answer to laplacetransform sin^3(t) is 6/((s^2+9)(s^2+1))