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Math Help - Fairly simple integration by parts query

  1. #1
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    Fairly simple integration by parts query

    Hi all!

    So, I have a question regarding integration by parts, and I think it's pretty straight-forward, but it's bugging the hell out of me...

    Supposing I have an integration \int_{x_1}^{x_2}f(x)g(x)dx, and integrate by parts, I'd have \left[f(x)\int g(x) dx\right]_{x_1}^{x_2} - \int_{x_1}^{x_2}\frac{df(x)}{dx}\int g(x)dxdx...

    But, should there not be a boundary on the \int g(x) dx terms? If so, what would they be? It just seems odd to me, but I'm probably overlooking something.
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  2. #2
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    Re: Fairly simple integration by parts query

    I would say that no limits are required, because what is required is the antiderivative of g. That is, just rewriting your equation for the purposes of annotation:

    \int_{x_{1}}^{x_{2}}f(x)\,g(x)\,dx=\Bigg[f(x)\underbrace{\int g(x)\,dx}_{\text{a function}}\Bigg]_{x_{1}}^{x_{2}}-\int_{x_{1}}^{x_{2}}\frac{df(x)}{dx}\underbrace{ \int g(x)\,dx}_{\text{a function}}\,dx.

    I think it helps to remember from where this formula comes: the product rule for derivatives:

    (fg)'=f'g+fg'\implies

    fg=\int f'g\,dx+\int fg'\,dx\implies

    \int fg'\,dx=fg-\int f'g\,dx.

    Now, the second step there, where I integrated, could have been definite integrals. Then it would have looked like this:

    [fg]_{x_{1}}^{x_{2}}=\int_{x_{1}}^{x_{2}}f'g\,dx+\int_  {x_{1}}^{x_{2}}fg'\,dx.

    But notice that both terms in both integrands are just functions. Now the way you've expressed integration by parts, you have my g' equals your g. But they're still just functions.

    Does that help?
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