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Math Help - need help in understanding a probably basic integration concept

  1. #1
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    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.
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  2. #2
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    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 \int_a^x f(t)dt or \int_a^x f(s)ds, etc. It doesn't matter what you call the variable inside the integral.

    For example,
    \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)
    \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)
    \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.
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  3. #3
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    Re: need help in understanding a probably basic integration concept

    Got it. Thank you.
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