What is laplacetransform e^{7t}cos^2(t) ?

The answer to laplacetransform e^{7t}cos^2(t) is 1/(2(s-7))+(s-7)/(2((s-7)^2+4))

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laplacetransform e^{7t}cos^2(t)

laplace transform

e^{7 t} cos^{2} (t)

\frac{1}{2 ( s - 7 )} + \frac{s - 7}{2 ( ( s - 7 ) ^{2} + 4 )}

Solution steps

L {e^{7 t} cos^{2} (t)}

Rewrite using trig identities

L {\frac{1}{2} e^{7 t} + \frac{1}{2} e^{7 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 {e^{7 t}} + \frac{1}{2} L {e^{7 t} cos (2 t)}

L {e^{7 t}} : \frac{1}{s - 7}

L {e^{7 t} cos (2 t)} : \frac{s - 7}{( s - 7 ) ^{2} + 4}

= \frac{1}{2} \cdot \frac{1}{s - 7} + \frac{1}{2} \cdot \frac{s - 7}{( s - 7 ) ^{2} + 4}

Refine

\frac{1}{2} \frac{1}{s - 7} + \frac{1}{2} \frac{s - 7}{( s - 7 ) ^{2} + 4} : \frac{1}{2 ( s - 7 )} + \frac{s - 7}{2 ( ( s - 7 ) ^{2} + 4 )}

= \frac{1}{2 ( s - 7 )} + \frac{s - 7}{2 ( ( s - 7 ) ^{2} + 4 )}

\frac{\partial}{\partial x} (cos (8 x) sin (3 y))

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

\int 4 cos (2 x) (sin (2 x) + 2)^{3} d x

\frac{d}{d x} (x^{2} (e^{x}))

h \to 0 lim (\frac{( 16 + h ) - 4}{h})

What is laplacetransform e^{7t}cos^2(t) ?
The answer to laplacetransform e^{7t}cos^2(t) is 1/(2(s-7))+(s-7)/(2((s-7)^2+4))