laplace transform of t^2+2t+3

The Laplace transform of

t^{2} + 2 t + 3

with respect to

t

is the integral of

\exp (- s t) \cdot (t^{2} + 2 t + 3)

from 0 to +∞

For s > 0, the integral converges. By linearity, apply

L (t^{k}) = \frac{k!}{s^{k + 1}}

to each monomial, for nonnegative integer k

The sum of the transforms gives:

3 \cdot \frac{0!}{s} + 2 \cdot \frac{1!}{s^{2}} + \frac{2!}{s^{3}}

⇥ⓘ The factorial of 0 is defined to be 1:

0!

1

■Calculate the factorial of a positive integer by multiplying all integers from 1 to that integer:

1!

1

2!

2

⇥

3 \cdot \frac{1}{s} + 2 \cdot \frac{1}{s^{2}} + \frac{2}{s^{3}}

ⓘ Multiplying a fraction by a number multiplies its numerator:

3 \cdot \frac{1}{s}

\frac{3 \cdot 1}{s}

\frac{3}{s}

2 \cdot \frac{1}{s^{2}}

\frac{2 \cdot 1}{s^{2}}

\frac{2}{s^{2}}

▶Laplace transform:

\frac{3}{s} + \frac{2}{s^{2}} + \frac{2}{s^{3}}

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