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Homework Help: Common Integrals

  1. Jan 1, 2006 #1
    my book integrates this using "the standard integral"

    [tex]\int e^{at} cos \omega t dt = \frac{1}{a^2+ \omega^2} e^{at} (a cos \omega t+ \omega sin \omega t) +c [/tex]
    where [tex]a[/tex] is a constant

    what is the standard integral?
     
  2. jcsd
  3. Jan 1, 2006 #2

    hotvette

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    Homework Helper

    I'm guessing a bit, but I believe the term "standard integral" just means common, known, ones. For example, following would be considered a standard integral:

    [tex]\int x^n dx = \frac{x^{n+1}}{n+1} +c [/tex]
     
  4. Jan 1, 2006 #3

    Tx

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    I would say the 'standard integral' that your text is referring to would be Integration By Parts. Note: I am assuming 'w' is constant.

    If you have never seen the formula here it is:
    http://mathworld.wolfram.com/IntegrationbyParts.html
     
  5. Jan 1, 2006 #4
    ohh... wow, I feel dumb. okay, I thought it was going to be some weird trig arctan integral that I have never seen. by parts would do it. thank you!
     
  6. Jan 2, 2006 #5
    I always thought that standard integrals are more general results which can be utilised by plugging in values specific to your problem. In this case, your integral looks like a fairly general one to me.

    Anyway an alternative to integration by parts is the following:

    [tex]
    \int {e^{\left( {a + \omega i} \right)t} } dt
    [/tex]

    [tex] = \frac{1}{{a + \omega i}}e^{\left( {a + \omega i} \right)t} [/tex]

    [tex]
    = \frac{{a - \omega i}}{{a^2 + \omega ^2 }}e^{\left( {a + \omega i} \right)t}
    [/tex]

    [tex]
    = \frac{{a - \omega i}}{{a^2 + \omega ^2 }}e^{at} \left( {\cos \left( {\omega t} \right) + i\sin \left( {\omega t} \right)} \right)
    [/tex]

    [tex]
    \int {e^{at} \cos \left( {\omega t} \right)} dt = {\mathop{\rm Re}\nolimits} \left\{ I \right\}
    [/tex]

    [tex]
    = \frac{1}{{a^2 + \omega ^2 }}e^{at} \left( {a\cos \left( {\omega t} \right) + \omega \sin \left( {\omega t} \right)} \right)
    [/tex]

    I left out the constant of integration.
     
    Last edited: Jan 2, 2006
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