What is laplacetransform cos^2(pit) ?

The answer to laplacetransform cos^2(pit) is 1/(2s)+s/(2(s^2+4pi^2))

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laplacetransform cos^2(pit)

laplace transform

cos^{2} (π t)

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

Solution steps

L {cos^{2} (π t)}

Use the following identity:

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

= L {\frac{1}{2} + cos (2 π 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 π t)}

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

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

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

Refine

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

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

x^{2} y^{^{'}} + x y = y^{2}

x \to - 1 lim ((x + 1) \cdot ln (x + 1))

\frac{2}{s + 1} + 32 \cdot \frac{s}{s ^{2} + 9}

f (x) = x^{4}, (- 1, 1)

y = 4 x - 2 x^{2}

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