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Thread: Integration

  1. #1
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    Integration

    Integral (e^x)cosx dx

    Not sure how to start?

    Thanks.
    Last edited by Erghhh; Mar 10th 2009 at 01:47 PM. Reason: might've caused confusion.
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  2. #2
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    Quote Originally Posted by Erghhh View Post
    Integral e^xcosx dx

    Not sure how to start?

    Thanks.
    Take the DERIVATIVE twice, and you should find you get a nice equation which you can solve for \int{e^x\cos{x}\,dx}.

    Or use Integration by Parts twice.
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  3. #3
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    so i got integral -2(e^x)sinx dx?

    Haven't used this method of integration. Still unsure of what to do.
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    Quote Originally Posted by Erghhh View Post
    Integral (e^x)cosx dx

    Not sure how to start?

    Thanks.
    \frac{d}{dx}\left[e^x\cos{x}\right] = e^x\cos{x} - e^x\sin{x}

    So \int{e^x\cos{x} - e^x\sin{x}\,dx} = e^x\cos{x} + C

    \int{e^x\cos{x}\,dx} - \int{e^x\sin{x}\,dx} = e^x\cos{x} + C. Call this equation 1.

    Now we need to find \int{e^x\sin{x}\,dx}. We use the exact same method.

    \frac{d}{dx}\left[e^x\sin{x}\,dx\right] = e^x\sin{x} + e^x\cos{x}

    So \int{e^x\sin{x} + e^x\cos{x}\,dx} = e^x\sin{x}

    \int{e^x\sin{x}\,dx} + \int{e^x\cos{x}\,dx} = e^x\sin{x}

    \int{e^x\sin{x}\,dx} = e^x\sin{x} - \int{e^x\cos{x}\,dx}.

    Substitute this back into equation 1.


    \int{e^x\cos{x}\,dx} - \left[e^x\sin{x} - \int{e^x\cos{x}\,dx}\right] = e^x\cos{x} + C

    2\int{e^x\cos{x}\,dx} - e^x\sin{x} = e^x\cos{x} + C

    2\int{e^x\cos{x}\,dx} = e^x\sin{x} + e^x\cos{x} + C

    \int{e^x\cos{x}\,dx} = \frac{1}{2}e^x(\sin{x} + \cos{x}) + C.


    Note, C can take on any arbitrary constant, but its actual value will change throughout the integration... I've just used the same symbol each time because I'm lazy...
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