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Math Help - integration by parts

  1. #1
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    integration by parts

    \int e^{sin(x)}cos(x) dx

    So I do;
    u = e^{sin(x)}
    du = cos e^{sin(x)}
    dv = cos(x)
    v = sin(x)
    <br />
sin(x)e^{sin(x)} - \int sin(x)cose^{sin(x)}

    How do I proceed? It didn't become any simpler after using integration by parts.
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  2. #2
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    A simple sub, not parts. Just in case a picture helps...



    ... where



    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    The general drift is...


    And the rest...
    Spoiler:

    _________________________________________
    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
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  3. #3
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    Thank you for posting, but it only really confused me even more.
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  4. #4
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Archduke01 View Post
    \int e^{sin(x)}cos(x) dx

    So I do;
    u = e^{sin(x)}
    du = cos e^{sin(x)}
    dv = cos(x)
    v = sin(x)
    <br />
sin(x)e^{sin(x)} - \int sin(x)cose^{sin(x)}

    How do I proceed? It didn't become any simpler after using integration by parts.
    Why use integration by parts when it can be done conveniently with substiution?

    Let u = sinx

    you can do the rest!

    PS: Does your question specifically tell you to integrate by parts or does it just tell you to integrate the given function? Substitution method is very easy to use in this case
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  5. #5
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    Quote Originally Posted by harish21 View Post
    Why use integration by parts when it can be done conveniently with substiution?

    Let u = sinx

    you can do the rest!
    After everything I end up with

    \frac {e^{sin x}}{cos^2x}

    Can someone please tell me if this is right? Because the problem is actually one on a definite integral and the answer I get after plugging the numbers is zero, and I don't think that's right...

    b = pi
    a = 0

    You don't have to check with the numbers though, it's enough if you just tell me whether the below is correct;

    \frac {e^{sin x}}{cos^2x}
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  6. #6
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Archduke01 View Post
    After everything I end up with

    \frac {e^{sin x}}{cos^2x}

    Can someone please tell me if this is right? Because the problem is actually one on a definite integral and the answer I get after plugging the numbers is zero, and I don't think that's right...

    b = pi
    a = 0

    You don't have to check with the numbers though, it's enough if you just tell me whether the below is correct;

    \frac {e^{sin x}}{cos^2x}
    NO.

    let u = sinx, then du = cosx dx \implies dx = \frac{du}{cosx}

    and your integrand becomes:

    \int e^u \times cosx \frac{du}{cosx} = \int e^u du

    do you get it now?
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  7. #7
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    Smile

    Quote Originally Posted by Archduke01 View Post
    \int e^{sin(x)}cos(x) dx

    So I do;
    u = e^{sin(x)}
    du = cos e^{sin(x)}
    dv = cos(x)
    v = sin(x)
    <br />
sin(x)e^{sin(x)} - \int sin(x)cose^{sin(x)}

    How do I proceed? It didn't become any simpler after using integration by parts.
    you should know that a primitive of the following function e^{u(x)}u'(x) is just e^{u(x)}
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