What is laplacetransform (t-6.8)sin(6.2t) ?

The answer to laplacetransform (t-6.8)sin(6.2t) is (12.4s)/((s^2+38.44)^2)-(42.16)/(s^2+38.44)

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laplacetransform (t-6.8)sin(6.2t)

laplace transform

(t - 6.8) sin (6.2 t)

\frac{12.4 s}{( s ^{2} + 38.44 ) ^{2}} - \frac{42.16}{s ^{2} + 38.44}

Solution steps

L {sin (6.2 t) (t - 6.8)}

Expand

(t - 6.8) sin (6.2 t) : t sin (6.2 t) - 6.8 sin (6.2 t)

= L {t sin (6.2 t) - 6.8 sin (6.2 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)}

= L {t sin (6.2 t)} - 6.8 L {sin (6.2 t)}

L {t sin (6.2 t)} : \frac{12.4 s}{( s ^{2} + 38.44 ) ^{2}}

L {sin (6.2 t)} : \frac{6.2}{s ^{2} + 38.44}

= \frac{12.4 s}{( s ^{2} + 38.44 ) ^{2}} - 6.8 \cdot \frac{6.2}{s ^{2} + 38.44}

Refine

\frac{12.4 s}{( s ^{2} + 38.44 ) ^{2}} - 6.8 \frac{6.2}{s ^{2} + 38.44} : \frac{12.4 s}{( s ^{2} + 38.44 ) ^{2}} - \frac{42.16}{s ^{2} + 38.44}

= \frac{12.4 s}{( s ^{2} + 38.44 ) ^{2}} - \frac{42.16}{s ^{2} + 38.44}

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What is laplacetransform (t-6.8)sin(6.2t) ?
The answer to laplacetransform (t-6.8)sin(6.2t) is (12.4s)/((s^2+38.44)^2)-(42.16)/(s^2+38.44)