What is laplacetransform t^4+3e^{3t}t^2+e^{-t}t+1 ?

The answer to laplacetransform t^4+3e^{3t}t^2+e^{-t}t+1 is (24)/(s^5)+6/((s-3)^3)+1/((s+1)^2)+1/s

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laplacetransform t^4+3e^{3t}t^2+e^{-t}t+1

laplace transform

t^{4} + 3 e^{3 t} t^{2} + e^{- t} t + 1

\frac{24}{s ^{5}} + \frac{6}{( s - 3 ) ^{3}} + \frac{1}{( s + 1 ) ^{2}} + \frac{1}{s}

Solution steps

L {t^{4} + 3 e^{3 t} t^{2} + e^{- t} t + 1}

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 {t^{4}} + 3 L {e^{3 t} t^{2}} + L {e^{- t} t} + L {1}

L {t^{4}} : \frac{24}{s ^{5}}

L {e^{3 t} t^{2}} : \frac{2}{( s - 3 ) ^{3}}

L {e^{- t} t} : \frac{1}{( s + 1 ) ^{2}}

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

= \frac{24}{s ^{5}} + 3 \cdot \frac{2}{( s - 3 ) ^{3}} + \frac{1}{( s + 1 ) ^{2}} + \frac{1}{s}

Refine

\frac{24}{s ^{5}} + 3 \frac{2}{( s - 3 ) ^{3}} + \frac{1}{( s + 1 ) ^{2}} + \frac{1}{s} : \frac{24}{s ^{5}} + \frac{6}{( s - 3 ) ^{3}} + \frac{1}{( s + 1 ) ^{2}} + \frac{1}{s}

= \frac{24}{s ^{5}} + \frac{6}{( s - 3 ) ^{3}} + \frac{1}{( s + 1 ) ^{2}} + \frac{1}{s}

- \frac{4 x}{x ^{2} + 1}

x \to \infty lim (x + 8)

\frac{d}{d x} (8 x^{3} e^{- x})

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

\int \frac{x}{1 + x x} d x

What is laplacetransform t^4+3e^{3t}t^2+e^{-t}t+1 ?
The answer to laplacetransform t^4+3e^{3t}t^2+e^{-t}t+1 is (24)/(s^5)+6/((s-3)^3)+1/((s+1)^2)+1/s