What is laplacetransform tsin^2(t) ?

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

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laplacetransform tsin^2(t)

laplace transform

t sin^{2} (t)

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

Solution steps

L {t sin^{2} (t)}

Rewrite using trig identities

L {\frac{1}{2} t - \frac{1}{2} t cos (2 t)}

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)}

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

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

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

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

Refine

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

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

y = \frac{x ^{3}}{x + 3}, (3, \frac{9}{2})

\int \frac{2 cos ( x )}{sin ^{3} ( x )} d x

x cos (x - 3)

\int 2 x (x^{2} + 5)^{6} d x

\int \frac{1}{( x + y + z + 1 ) ^{3}} d z

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