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Math Help - help with intergration by recogbition

  1. #1
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    help with intergration by recogbition

    G(x)=(10-x)cos((pi)x)
    find an expression for the derivative G'(x)
    i got that but then it says
    hence show that an antiderivative of f(x) is
    F(x)= ((sin((pi)x)) \ (pi)) - G(x)

    and f(x) =pi(10-x)sn(pi)x
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  2. #2
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    Quote Originally Posted by dustinbehemoth View Post
    G(x)=(10-x)cos((pi)x)
    find an expression for the derivative G'(x)
    i got that but then it says
    hence show that an antiderivative of f(x) is
    F(x)= ((sin((pi)x)) \ (pi)) - G(x)

    and f(x) =pi(10-x)sn(pi)x
    What do you get for the derivative?
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  3. #3
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    the derivative of g(x) was
    (pi)(10-x)sin(pi)x - cos(pi)x
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  4. #4
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    Quote Originally Posted by dustinbehemoth View Post
    the derivative of g(x) was
    (pi)(10-x)sin(pi)x - cos(pi)x
    Let y =(10-x) \cos ( \pi x) .... (1)

    Then \frac{dy}{dx} = \pi (10 - x) \sin (\pi x) - \cos (\pi x) .... (2)

    Integrate both sides of (2) with respect to x:

    y = \int  \pi (10 - x) \sin (\pi x) - \cos (\pi x) \, dx =  \int  \pi (10 - x) \sin (\pi x) \, dx - \int \cos (\pi x) \, dx .

    Substitute from (1):

    (10-x) \cos ( \pi x) = \int \pi (10 - x) \sin (\pi x) \, dx - \int \cos (\pi x) \, dx .

    Make the required integral the subject etc.
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  5. #5
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    found a heaps easier(may just be better explained) way but thanks anyway
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