What is laplacetransform cos^2(ax) ?

The answer to laplacetransform cos^2(ax) is 1/(2s)+s/(2(s^2+4a^2))

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laplacetransform cos^2(ax)

laplace transform

cos^{2} (a x)

\frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 4 a ^{2} )}

Solution steps

L {cos^{2} (a x)}

Use the following identity:

cos^{2} (x) = \frac{1}{2} + cos (2 x) \frac{1}{2}

= L {\frac{1}{2} + cos (2 a x) \frac{1}{2}}

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 (2 a x)}

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

L {cos (2 a x)} : \frac{s}{s ^{2} + 4 a ^{2}}

= \frac{1}{2 s} + \frac{1}{2} \cdot \frac{s}{s ^{2} + 4 a ^{2}}

Refine

\frac{1}{2 s} + \frac{1}{2} \frac{s}{s ^{2} + 4 a ^{2}} : \frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 4 a ^{2} )}

= \frac{1}{2 s} + \frac{s}{2 ( s ^{2} + 4 a ^{2} )}

f^{^{'}} (s) = 10 s - 12 s^{3}, f (3) = 2

n = 0 \sum \infty x^{2 n}

y = x^{2} + 1, (2, 5)

\frac{d}{d t} (e^{7 t})

\int \frac{1}{( x + 5 ) 10 x + x ^{2}} d x

What is laplacetransform cos^2(ax) ?
The answer to laplacetransform cos^2(ax) is 1/(2s)+s/(2(s^2+4a^2))