What is laplacetransform cos^2(t) ?

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

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laplacetransform cos^2(t)

laplace transform

cos^{2} (t)

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

Solution steps

L {cos^{2} (t)}

Use the following identity:

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

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

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

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

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

Refine

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

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

\frac{\partial}{\partial x} (7 e^{x y} \cdot cos (x yz))

x \to 0 lim (\frac{f ( x )}{x})

\frac{d y}{d x} = y (1000 - y)

y = 5 e^{- x}, (0, 5)

\frac{2}{( s ^{2} - 3 )} - 3 \frac{s}{( s ^{2} - 3 )}

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