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Math Help - Integrate e^2x sin x by parts [Check & Correct my working please]

  1. #1
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    Integrate e^2x sin x by parts [Check & Correct my working please]

    Hello

    I am having trouble with integrating by parts
    I am trying to integrate e^2x * sin x

    I have attached my working, and I am getting stuck towards the end. Please let me know where I am going wrong.

    Thank you
    http://img709.imageshack.us/img709/3985/20110609001.jpg
    Last edited by anees7112; June 8th 2011 at 09:41 AM.
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  2. #2
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    It seems you've made a substitution that reversed what you did earlier:

    \displaystyle \begin{aligned}\int f'(x)g(x)\;{dx} & = f(x)g(x)\;{dx}-\int f(x)g'(x)\;{dx} \\& = f(x)g(x)-f(x)g(x)+\int f'(x)g(x)\;{dx} \\& = \int f'(x)g(x)\;{dx}.\end{aligned}

    The above is a pitfall to beware of when doing integration by parts.
    I remember there was a similar thread a while back - see it in here.
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  3. #3
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    Thank you, but how would I go about fixing this?
    Edit: Okay, will have a look now
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  4. #4
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    Quote Originally Posted by anees7112 View Post
    Thank you, but how would I go about fixing this?
    Let u = e^{2x} again for the second integration by parts.
    By the way, try this method too (the same method really).
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  5. #5
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    Here is the problem I was having when I tried to integrate witu u as e^2x :
    http://img38.imageshack.us/img38/2369/20110609002.jpg

    And thanks for the link, I will look over it now
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  6. #6
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    Quote Originally Posted by anees7112 View Post
    One minute, still a small problem
    Are you sure? The middle terms shouldn't cancel, and the sign of the last integral should be negative, so that you can take it to the other side.
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  7. #7
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    I'm not sure if you saw it but I edited the last post, still stuck on something there it looks like .
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  8. #8
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    Quote Originally Posted by TheCoffeeMachine View Post
    By the way, try this method too (the same method really).

    And here's as far as the OP originally got right, in balloon sculpture.

    And filling the blanks (and solving the top row for I) does confirm his/her later answer, however arrived at.



    ... where (key in spoiler) ...

    Spoiler:


    ... is the product rule, straight continuous lines differentiating downwards with respect to x.



    ... is lazy integration by parts, doing without u and v.


    Larger


    __________________________________________________ __________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
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  9. #9
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    Haha I have never seen anything like that before, Tom! I would love to learn how the baloons work tonight, but I will have to postpone it until exams are over.

    I tried to use wolfram to confirm (1/3)*(e^(2x)*cos x +e^(2x) sin x)=(1/5)(2 sin x- cos x) but it wouldn't give the True or False answer it sometimes does.

    Anyhow, thank you again TheCoffeeMachine and Tom@balooncalculus very much Good night
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  10. #10
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    Quote Originally Posted by anees7112 View Post
    I'm not sure if you saw it but I edited the last post, still stuck on something there it looks like .
    You forgot the minus that you had before \cos{x}e^{2x} and (2e^{2x})' = 4e^{2x}.
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  11. #11
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    Quote Originally Posted by tom@ballooncalculus View Post
    is lazy integration by parts
    I love lazy integration by parts!
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  12. #12
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    Ah okay thank you so much for taking the time to show me how to do this . One last thing, so should my second u and v' always be the same half of the equation if you know what I mean?

    I let u= e^(2x) first but then for the second time I incorrectly let u=cos x instead of e^(2x) again... Is that what you meant I should be aware of when doing by parts?
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  13. #13
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    Quote Originally Posted by anees7112 View Post
    Ah okay thank you so much for taking the time to show me how to do this .
    You're welcome.
    One last thing, so should my second u and v' always be the same half of the equation if you know what I mean? I let u= e^(2x) first but then for the second time I incorrectly let u=cos x instead of e^(2x) again... Is that what you meant I should be aware of when doing by parts?
    Yes, basically making 'a substitution that precisely nullifies the effect of a given substitution'.
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  14. #14
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    Great! I have spent a lot of time stuck on this question and now I can finally move on Night Night
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  15. #15
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    Quote Originally Posted by anees7112 View Post
    Great! I have spent a lot of time stuck on this question and now I can finally move on Night Night
    Good night! I'm off too to celebrate my 7th post.
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