What is laplacetransform cos^2(kt) ?

The answer to laplacetransform cos^2(kt) is 1/(2s)+s/(2(s^2+4k^2))

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laplacetransform cos^2(kt)

laplace transform

cos^{2} (k t)

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

Solution steps

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

Use the following identity:

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

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

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

L {cos (2 k t)} : \frac{s}{s ^{2} + 4 k ^{2}}

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

Refine

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

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

\int_{0}^{3} 3 d x

\frac{d}{d x} (\frac{x - ln ( 1 + x )}{x ^{2}})

\int x ln (3) d x

x \to 1 + lim (\frac{2 - x}{x - 1})

y^{^{''}} - 12 y^{^{'}} + 36 y = 4 x e^{6 x}

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