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Thread: Integral involving an integral part

  1. #1
    Junior Member
    May 2012

    Question Integral involving an integral part

    I recently started working through Arfken, Weber and Harris' Mathematical Methods for Physicists, when I came across the following integral:

    $\displaystyle \int_{N_1}^{N_2} [x] f'(x) \, dx = \sum_{n=N_1}^{N_2-1} n \int_{n}^{n+1} f'(x)\, dx$

    where $\displaystyle [x]$ is the integral part of $\displaystyle x$.

    I don't understand how the RHS emerges from the LHS. Can someone please enlighten me as to what the hidden steps are? Thanks very much.

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  2. #2
    MHF Contributor

    Apr 2005

    Re: Integral involving an integral part

    It looks pretty straight forward to me. One of the very first things you learn about integrals is that "$\displaystyle \int_a^c g(x)dx= \int_a^b g(x)dx+ \int_b^c g(x)dx$". I presume you will agree that $\displaystyle \int_0^3 g(x)dx= \int_0^1 g(x)dx+ \int_1^2 g(x)dx+ \int_2^3 g(x) dx= \sum_{n= 0}^2 \int_n^{n+1} g(x)dx$. Extending that, the left side is equal to $\displaystyle \sum_{n=N_1}^{N_2- 1} \int_{n}^{n+1} [x]f'(x)dx$ since that just breaks the integral into a sum of integrals. If x lies between n and n+ 1 then [x]= n so that $\displaystyle \int_n^{n+1} [x]f'(x)dx= n\int_n^{n+1} f'(x)dx$.
    Last edited by HallsofIvy; Jun 11th 2014 at 11:10 AM.
    Thanks from britmath
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