What is laplacetransform sin^2(pit) ?

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

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laplacetransform sin^2(pit)

laplace transform

sin^{2} (π t)

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

Solution steps

L {sin^{2} (π t)}

Use the following identity:

sin^{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} )}

\int sin (\frac{π}{2}) d x

\int cot^{5} (x) \cdot csc^{3} (x) d x

8 \frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} + 9 y = 0

n = 0 \sum \infty ln (1 + \frac{1}{n})

y^{^{''}} - 10 y^{^{'}} + 34 y = 0, y (0) = 2, y^{^{'}} (0) = 3

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