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Math Help - Anti-differentiation

  1. #1
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    Anti-differentiation

    Hey guys, sorry if this question is posted in the wrong section but i didnt no where else to submit it. Anywayz, the question is:

    Find the anti-derivative of

    e^4x / e^x - 1

    Ive looked at the question, tried a partial fraction, tried the change of variable method but cant get close to the answer. Working out would be much appreciated. Thankz
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  2. #2
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    Try u = e^x - 1

    Try the "Calculus" Section
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  3. #3
    Moo
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    Hello,

    Quote Originally Posted by sanado View Post
    Hey guys, sorry if this question is posted in the wrong section but i didnt no where else to submit it. Anywayz, the question is:

    Find the anti-derivative of

    e^4x / e^x - 1

    Ive looked at the question, tried a partial fraction, tried the change of variable method but cant get close to the answer. Working out would be much appreciated. Thankz
    Hmm well, partial fraction would have worked...


    Note that x^4=(x-1)(x^3+x^2+x+1)+1

    Therefore \frac{e^{4x}}{e^x-1}=\frac{(e^x-1)(e^{3x}+e^{2x}+e^x+1)+1}{e^x-1}

    =e^{3x}+e^{2x}+e^x+1+\frac{1}{e^x-1}


    --> \int \frac{e^{4x}}{e^x-1} \, dx=\int e^{3x} \, dx+\int e^{2x} \, dx+\int e^x \, dx+\int \, dx+\int \frac{1}{e^x-1} \, dx

    For the last one, try u=e^x-1 and don't forget the constant of integration.
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  4. #4
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by Moo View Post
    Note that x^4=(x-1)(x^3+x^2+x+1)+1

    Therefore \frac{e^{4x}}{e^x-1}=\frac{(e^x-1)(e^{3x}+e^{2x}+e^x+1)+1}{e^x-1}

    =e^{3x}+e^{2x}+e^x+1+\frac{1}{e^x-1}
    @ sanado :

    To find this you can think about the sum of the terms of a geometric sequence : 1+u+u^2+\ldots+u^n=\frac{1-u^{n+1}}{1-u}

    Letting n=3 and u=\exp x it gives 1+\exp x +\exp (2x)+\exp(3x)=\frac{1-\exp (4x)}{1-\exp(x)} and from this you can easily find Moo's result : \frac{\exp(4x)}{\exp x-1}=1+\exp x +\exp (2x)+\exp(3x)+\frac{1}{\exp x-1}
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  5. #5
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    Hmmmmm ive tried the method above but i aint getting the answer which is:

    1/3.(e^x - 1)^3 + 3/2.(e^x - 1)^2 + 3(e^x - 1) + loge(e^x - 1) + c
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  6. #6
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by sanado View Post
    Hmmmmm ive tried the method above but i aint getting the answer which is:

    1/3.(e^x - 1)^3 + 3/2.(e^x - 1)^2 + 3(e^x - 1) + loge(e^x - 1) + c
    The method above should give you \ln(\exp x-1)+\frac{\exp(3x)}{3}+\frac{\exp(2x)}{2}+\exp x+C which is the same as the answer. (hopefuly ) If it still does not correspond to what you've found, show us what you've done so that we can try to find what the problem is.
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  7. #7
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    Wow! That's a lot of work.

    \int \frac{e^{4x}}{e^{x}-1}\;dx

    u = e^{x}-1

    du = e^{x}dx

    u+1 = e^{x}

    \int \frac{(u+1)^{3}}{u}\;du

    It's a simple polynomial.
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