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Easy integral

  1. Jul 6, 2008 #1
    [tex]
    \int \frac{x \cos(x) - \sin(x)}{( x-\sin(x))^2}\ \mbox{d}x
    [/tex]

    I don't see which substitution I should use can anyone help me?
     
  2. jcsd
  3. Jul 6, 2008 #2

    morphism

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    The integrand looks a lot like something you'd get if you differentiated an expression of the form f(x)/g(x) using the quotient rule.
     
  4. Jul 7, 2008 #3
    That I know, but it isn't helping me. I'm interested in a structured manner of solving this problem.
     
  5. Jul 7, 2008 #4

    matt grime

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    So guessing a substitution is structured, but noting the form of the integrand is 'unstructured'? That seems like a highly arbitrary choice to make. Since integrals are generically impossible to do by hand, I'd take what you can get when you can get it.
     
  6. Jul 7, 2008 #5
    Indeed, with morphism's observation, a keen eye, and some algebra, you can easily arrive at the solution without any significant integration. But I have no other useful suggestion if your intent is otherwise.
     
  7. Jul 7, 2008 #6
    Fine. So how do I proceed next? I'm seeing something of a quotient rule but how can you find the primitive?
     
  8. Jul 7, 2008 #7

    Dick

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    Just guess. Write down the form of (f(x)/g(x))'. What's a good guess for g(x)? Put that in. That's easy. Now start hunting for an f(x) that works.
     
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