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Thread: Substitution rule for integrals

  1. #1
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    Question Substitution rule for integrals

    Hi,

    I hope I can get some help with solving a problem.

    The question is as follows: Substitution rule for integrals-screen-shot-2019-07-21-10.43.26-am.png

    Intuitively, I know the answer is 0.

    Now I am in the process of proving it and I'm stuck.

    Here is the work that I have done so far:

    Substitution rule for integrals-img_20190721_104027.jpg

    What I am confused about: do you interpret the problem as though f has an inner function, and that inner function being x? And if so, it's still not clear to me why you are "allowed" to assign u (the substitution variable) to - x. I see how this assignment is helpful towards solving the problem, but I don't see how this is allowed to be done in the first place, given my current understanding of the substitution rule. I see how assigning u to x would be valid, but not -x.

    Help would be appreciated!
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  2. #2
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    Re: Substitution rule for integrals

    $I=\displaystyle \int_{-a}^0 f(x)~dx = \int_{-a}^0 -f(-x)~dx$

    Now the $-x$ appears explicitly in the integral. This may make it easier for you to accept

    $u=-x,~du=-dx$

    $I= \displaystyle \int_a^0 f(u)du = -\int_0^a f(u)~du = -\int_0^a f(x)~dx$

    The last equality is because $u$ and $x$ are just dummy variables in the integral so we can call them whatever we like.
    Thanks from otownsend
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  3. #3
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    Re: Substitution rule for integrals

    Thank you so much!
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