Laplace Transform Question (Diff Eqn)?

Question

Stuck on some homework and I have a test Wednesday!

F(s)=L{f(t)}
f(t) is a piecewise continuous and of exponential order [0,infinity)

I need to show that...
L{(integral from 0 to t)f(x)dx}=(1/s)F(s)

I'm given a hint to let g(t)=(integral from 0-infinity)f(t))dt and that g'(t)=f(t). Then I'm told to use L{f'(t)}=sL{f(t)}-f(0).

Any help is greatly appreciated! Thanks!

Answer 1

I'm not sure what prior Laplace relationships you are allowed to use, but if you set g(t)=∫f(t)dt, so g'(t)=f(t). Now take L transform of both sides to get L{∫f(t)dt} = L{g(t)}. But L{g'(t)} = s*L{g(t)} - g(0); then L{g'(t)} = s*L{g(t)}-g(0) g'(t) = f(t) L{f(t)} = s*L{g(t) - g(0)

[L{f(t)}+f'(0)]/s = L{g(t)}

L{g(t)} = L{∫f(t)dt}

L(∫f(t)dt} = 1/s * L{f(t)} + f'(0)/s

You need to show that f'(0) = 0

Answer 2

http://www.sosmath.com/diffeq/laplace/basic/basic.

http://images.google.com/imgres?imgurl=http://mo.mathematik.uni-stuttgart.de/inhalt/beispiel/beispiel686/img3.png&imgrefurl=http://mo.mathematik.uni-stuttgart.de/kurse/kurs16/seite59.html&h=254&w=309&sz=4&hl=en&start=1&tbnid=kx70gebQ4kBgDM:&tbnh=96&tbnw=117&prev=/images%3Fq%3Dlaplace%2Btransformation%26svnum%3D10%26hl%3Den%26lr%3D%26client%3Dfirefox-a%26rls%3Dorg.mozilla:en-US:official%26hs%3DKL0%26sa%3DX

look at this picture, try to understand how the math respresents
the curve in the graph

http://www.intmath.com/Laplace/6_lap_laptransint.php
http://www.intmath.com/Laplace/9_lap_invlaptrans_intDE.php
http://www.intmath.com/Laplace/2_lap_defn.php

http://www.intmath.com/Laplace/7_lap_invlaptrans.php
http://www.intmath.com/Laplace/8_lap_invlaptrans_DE.php

look thru this website, it has all the answers worked out