A helpful tip for integration by parts

Discussion in 'Math' started by amilton542, Oct 21, 2014.

  1. amilton542

    Thread Starter Active Member

    Nov 13, 2010
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    64
    There comes a time and a place when a difficult integration can become an easy one, i.e.

     F(x) = \int u\ dv = uv - \int v\;du

    Consider,

     F(x) = \int\;x^9cos\;ax\;dx

    You will need to perform integration by parts TEN times in order to reduce the variable "x" and its exponent to a constant. This is (by hand) horrendous computation.

    Let's integrate,

     G(x) = \int\; x^3\;cos\;ax\;dx

    So,

     G(x) = \frac{x^3\;sin\;ax}{a} + \frac{(3x^2)\;cos\;ax}{a^2} - \frac{(6x)\;sin\;ax}{a^3} - \frac{(6)cos\;ax}{a^4} + C

    Can you see the pattern?

    The first term is uv, for each term there after, u is differentiated, v is integrated and the result is multiplied by minus 1 for obvious reasons in the integration by parts formula.

    Given any product of the form,

     F(x) = \int\;(a_nx^n + a_{n-1} x^{n-1}+... + a_{0}x^{0})cos\;ax\;dx

    Where v can be a sine or cosine, can be integrated in less than one minute, provided the exponent is within reason of course.

    Let's return to the first example

     F(x) = \int\;x^9cos\;ax\;dx

    and now use the method claimed.

    You will be able to integrate this (by hand) in less than one minute I assure you.
     
    Last edited: Oct 21, 2014
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