What is laplacetransform cos^2(3t) ?

The answer to laplacetransform cos^2(3t) is 1/(2s)+s/(2(s^2+36))

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laplacetransform cos^2(3t)

laplace transform

cos^{2} (3 t)

\frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 36 )}

Solution steps

L {cos^{2} (3 t)}

Rewrite using trig identities

L {\frac{1}{2} + \frac{1}{2} cos (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 {\frac{1}{2}} + \frac{1}{2} L {cos (6 t)}

L {\frac{1}{2}} : \frac{1}{2 s}

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

= \frac{1}{2 s} + \frac{1}{2} \cdot \frac{s}{s ^{2} + 36}

Refine

\frac{1}{2 s} + \frac{1}{2} \frac{s}{s ^{2} + 36} : \frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 36 )}

= \frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 36 )}

y^{^{''}} + 2 y^{^{'}} + 2 y = sin (2 t)

sin^{5} (4 x^{5})

\int sin (2 x) (sin^{2} (x) + π)^{\frac{3}{2}} d x

f (x) = e^{s i n (x)}

What is laplacetransform cos^2(3t) ?
The answer to laplacetransform cos^2(3t) is 1/(2s)+s/(2(s^2+36))