need help in understanding a probably basic integration concept

I was reading a presentation on Ito's process. I read the following.

dx = a(x,t)dt

x(t) = x(0) + intgration from 0 to t (a(x,s))ds

What I do not understand is from where did 's' come and what is 's'? I think I have forgotten the basics of integration.

Thank you for your help.

Re: need help in understanding a probably basic integration concept

The s is a "dummy" variable. If you have function f(x) any anti-derivative is $\displaystyle \int_a^x f(t)dt$ or $\displaystyle \int_a^x f(s)ds$, etc. It doesn't matter what you call the variable inside the integral.

For example,

$\displaystyle \int_1^x t^2 dt= \left[\frac{1}{3}t^3\right]_1^x= \frac{1}{3}x^3- \frac{1}{3}(1)^3= \frac{1}{3}(x^3- 1)$

$\displaystyle \int_1^x s^2 ds= \left[\frac{1}{3}s^3\right]_1^x= \frac{1}{3}x^3- \frac{1}{3}(1)^3= \frac{1}{3}(x^3- 1)$

$\displaystyle \int_1^x u^2 du= \left[\frac{1}{3}u^3\right]_1^x= \frac{1}{3}x^3- \frac{1}{3}(1)^3= \frac{1}{3}(x^3- 1)$

They are all the same.

Re: need help in understanding a probably basic integration concept