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Math Help - [SOLVED] Integration by parts and trig identity help

  1. #1
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    [SOLVED] Integration by parts and trig identity help

    <br />
\int xcos^2(2x)~dx<br />

    I can tell that this needs some sort of trig identity in order to solve, but I may be wrong.

    How would I set this problem up with regards to u and dv?
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  2. #2
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    \cos^2{2x} = \frac{1 + \cos{4x}}{2}

    Try it now. Choose u and dv in a way so that the next integral would only have Sine.
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  3. #3
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    So how would that problem look if it was multiplied out? Or should I not do that?

    <br />
\int x*\frac {1+cos4x}{2}~dx = \int \frac {x+cos4x^2}{2}~dx<br />

    Because that looks confusing and I don't know what to do next.
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  4. #4
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    You cannot multiply x with the inside terms of cosine!

    \int x\cos^2(2x)~dx

    = \int x(\frac{1+\cos{4x}}{2})

    = \int \frac{x+x\cos{4x}}{2}

    Now split.

    = \int \frac{x}{2} + \int \frac{x\cos{4x}}{2}

    I presume you know what to do next.
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