What is laplacetransform (t-(3pi)/6)sin(6t) ?

The answer to laplacetransform (t-(3pi)/6)sin(6t) is (12s)/((s^2+36)^2)-(3pi)/(s^2+36)

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laplacetransform (t-(3pi)/6)sin(6t)

laplace transform

(t - \frac{3 π}{6}) sin (6 t)

\frac{12 s}{( s ^{2} + 36 ) ^{2}} - \frac{3 π}{s ^{2} + 36}

Solution steps

L {sin (6 t) (t - \frac{3 π}{6})}

Expand

(t - \frac{3 π}{6}) sin (6 t) : t sin (6 t) - \frac{π}{2} sin (6 t)

= L {t sin (6 t) - \frac{π}{2} sin (6 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 t)} - \frac{π}{2} L {sin (6 t)}

L {t sin (6 t)} : \frac{12 s}{( s ^{2} + 36 ) ^{2}}

L {sin (6 t)} : \frac{6}{s ^{2} + 36}

= \frac{12 s}{( s ^{2} + 36 ) ^{2}} - \frac{π}{2} \cdot \frac{6}{s ^{2} + 36}

Refine

\frac{12 s}{( s ^{2} + 36 ) ^{2}} - \frac{π}{2} \frac{6}{s ^{2} + 36} : \frac{12 s}{( s ^{2} + 36 ) ^{2}} - \frac{3 π}{s ^{2} + 36}

= \frac{12 s}{( s ^{2} + 36 ) ^{2}} - \frac{3 π}{s ^{2} + 36}

\frac{d}{d x} (6 ln (cos (x)))

\frac{\partial}{\partial x} (e^{x y} sin (y))

x \to 0 lim (9 x^{2} - 2 x + 5)

\frac{\partial}{\partial b} (1 0^{- 2 b})

\int \frac{x ^{2}}{x ^{2} + x - 6} d x

What is laplacetransform (t-(3pi)/6)sin(6t) ?
The answer to laplacetransform (t-(3pi)/6)sin(6t) is (12s)/((s^2+36)^2)-(3pi)/(s^2+36)