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Math Help - Integration By Parts...e^x * cosx dx

  1. #1
    Senior Member nycmath's Avatar
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    Integration By Parts...e^x * cosx dx

    Let INT = integral symbol


    INT( e^x)(cosx) dx


    What is u and dv here to get me started?
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  2. #2
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    Re: Integration By Parts...e^x * cosx dx

    use ILATE LAW
    if u=first fn(higher precedence) which here is of cos(x)
    v=second fn IS e^(x)
    now
    I=COS(X)INTEGRATE(E^(X))-INTEGRATE(DIFF(COS(X)INTEGRATE(E^(X)))
    I=COS(X)E^(X)+INTEGRATE(SIN(X)E^(X))
    NOW APPLY INTEGRATION OF PARTS ONE MORE TIME
    I=COS(X)E^(X)+E^(X)SIN(X)-I WHERE I=INTEGRATE(E^(X)COS(X)) [UNDERSTAND IN TERMS OF RECURSION]
    SOLVING I U GET
    I=E^(X)(COS(X)+SIN(X))/2+CONSTANT
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  3. #3
    Senior Member nycmath's Avatar
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    Re: Integration By Parts...e^x * cosx dx

    Quote Originally Posted by prasum View Post
    use ILATE LAW
    if u=first fn(higher precedence) which here is of cos(x)
    v=second fn IS e^(x)
    now
    I=COS(X)INTEGRATE(E^(X))-INTEGRATE(DIFF(COS(X)INTEGRATE(E^(X)))
    I=COS(X)E^(X)+INTEGRATE(SIN(X)E^(X))
    NOW APPLY INTEGRATION OF PARTS ONE MORE TIME
    I=COS(X)E^(X)+E^(X)SIN(X)-I WHERE I=INTEGRATE(E^(X)COS(X)) [UNDERSTAND IN TERMS OF RECURSION]
    SOLVING I U GET
    I=E^(X)(COS(X)+SIN(X))/2+CONSTANT
    Sorry but your reply is hard to read. Can you reply using latex?
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    Re: Integration By Parts...e^x * cosx dx

    Quote Originally Posted by nycmath View Post
    Let INT = integral symbol


    INT( e^x)(cosx) dx


    What is u and dv here to get me started?
    Either way will work:
    a) Let u= e^x and dv= cos(x) dx. Then du= e^x dx and v= sin(x) so we have \int e^x cos(x)dx= uv- \int v du= (e^x sin(x))- \int e^x sin(x) dx. To integrate \int e^x sin(x), let u= e^x and dv= sin(x)dx. Then du= e^x dx and v= -cos(x) so we have \int e^x sin(x)dx= uv- \int vdu= -e^x cos(x)+ \int e^x cos(x)dx.

    Putting those together,
    \int e^x cos(x)dx= e^x sin(x)- (-e^x cos(x)+ e^x cos(x)dx)= e^x(sin(x)+ cos(x))- \int e^x cos(x)dx
    and then 2\int e^x cos(x)dx= e^x(sin(x)+ cos(x)),
    \int e^x cos(x)dx= \frac{1}{2}e^{x}(sin(x)+ cos(x))+ C.

    Now, you try it letting u= cos(x), dv= e^x dx.

    (By the way- when you are not sure which of two functions to take as "u" and which to take as "dv", try both!)
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    Re: Integration By Parts...e^x * cosx dx

    You don't use LaTeX, but you want prasum to?

    - Hollywood
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  6. #6
    Senior Member nycmath's Avatar
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    Re: Integration By Parts...e^x * cosx dx

    Quote Originally Posted by hollywood View Post
    You don't use LaTeX, but you want prasum to?

    - Hollywood
    I never learned how to use LaTeX.
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    Re: Integration By Parts...e^x * cosx dx

    Quote Originally Posted by nycmath View Post
    I never learned how to use LaTeX.
    You should never ask someone to do something you are not willing to do yourself. We have a LaTeX subforum with tutorials, not to mention the world at our fingertips with Google, so learn it!
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  8. #8
    Senior Member nycmath's Avatar
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    Re: Integration By Parts...e^x * cosx dx

    Quote Originally Posted by Prove It View Post
    You should never ask someone to do something you are not willing to do yourself. We have a LaTeX subforum with tutorials, not to mention the world at our fingertips with Google, so learn it!

    Right now my concern is calculus.
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  9. #9
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    Re: Integration By Parts...e^x * cosx dx

    You seem to be missing the point- whatever excuse (reason) you have for not using Latex, prasum, or any other person here, may have the same reason. Who are to ask that other people do what you, for whatever reason, do not do?
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