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How do you know when to use integration by parts on a problem?

  1. Sep 23, 2009 #1
    This a techniques of integration question, and I'm wondering how do you know when to use integration by parts on a problem?

    My book says this bout the Integration by parts procedure. If f(x) is a product of a power of x and transcendental function then we try integration by parts.

    Can someone please show me examples of problems?
    Last edited: Sep 23, 2009
  2. jcsd
  3. Sep 23, 2009 #2


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    Integration is kind of like black magic. There are times when it's obvious, for example, whenever you have a polynomial multiplied by something you know how to integrate, just keep differentiating the polynomial until it goes away

    Or if you have something multiplied by cosine or sine, you can usually differentiate the something and integrate the trig function twice to get a formula for the integral in terms of itself (which can then be solved).

    Whenever you want to prove a recursive relationship about integrals involving a parameter of a natural number somewhere, you integrate by parts.

    There's no real tried and true method for determining how to find an integral, a lot of it is just practice and familiarity with which kinds of techniques work when
  4. Sep 23, 2009 #3
    Thanks I think I have a better understanding.
  5. Sep 25, 2009 #4
    Do every problem in your book and you will get a better feel for when it is useful. When the derivative of a function is easier to deal with than the original function, it is useful.
  6. Sep 29, 2009 #5
    -Suppose you want to integrate something like arcsinx* x/Sqrt(1-x^2)dx
    1) recognise that x/Sqrt(1-x^2) is in fact derivative of -Sqrt(1-x^2)
    2) Then You have in the integral -arcsinx*d(Sqrt(1-x^2)dx
    3)Using partial integration you then have
    -arcsinx*Sqrt(1-x^2)+integral Sqrt(1-x^2)/Sqrt(1-x^2)dx=-arcsinx*Sqrt(1-x^2)+x + C
    I believe that in this example you can see the beaty and the advantage of partial integration:)
    -Integrate ln(x)dx=xlnx-integral(x/x)dx=xlnx-x+C
    Here you must use integration by parts because you cannot directly integrate ln(x).
    The other common use would be your definition example.
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