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Math Help - When to use each method of integration?

  1. #1
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    When to use each method of integration?

    My problem for a while has been deciding which method of integration to use in every given situation.

    Are there times when you could use either Integration by Parts or U-substitution to achieve the same answer? Is this always/never the case?


    In what situations do you know for sure to use each of these methods?
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  2. #2
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    Re: When to use each method of integration?

    It's a matter of experience, and being able to "see" how to do it. I can think of integrals where multiple substitutions all work; it wouldn't surprise me if there are integrals that succumb to multiple methods. Here's an example:

    \int_{0}^{10}(-2x)e^{-x^{2}}\,dx.

    I could substitute either u=e^{-x^{2}} or u=-x^{2}, and I'd get the same result either way, as I should.
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  3. #3
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    Re: When to use each method of integration?

    My usual way is to try the simplest thing I think will work and if it doesn't, try something else!
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: When to use each method of integration?

    You have a lot of integration techniques but in the first place it's important I think to look if you're integrand is:
    - rational function (often: achieve the euclidian division, split in partial fractions, ...)
    - a product of two functions (often: use integration by parts, ...)
    - irrational function (often: goniometric substitution, ...)
    - trigoniometric function (often: formulas of Simpson, using t-formulas, ... -> good acknowledge of goniometric formulas)
    - ...

    But offcourse the most important thing is that you can use substitution.
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