Find the laplace transform of a constant

Answer 1

Let f(x) = 1

Then the laplace transform of 1 is

L[1] = ∫₀^∞ e^-px f(x) dx

then the right hand side becomes

= ∫₀^∞ e^-px 1 dx

= ∫₀^∞ e^-px dx

Now find the integral

= [ (-e^-px ⁄ p)]₀^∞ + C

= substituting the intervals

= [ (-e^-p∞ ⁄ p) - (-e^-p0 ⁄ p)]

= [ (-e^-∞ ⁄ p) - (-e^-0 ⁄ p)]

Since ∞ is in the denominator, it drives the fraction towards 0

= [ 0 + e⁰ ⁄ p]

= [ 1/p]

So, the laplace transform of 1 is 1/p

the lapalce transform of 2 is 2/p

and so on.