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Math Help - splitting integrals!

  1. #1
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    splitting integrals!

    i need some help on how to split integrals. for instance this

     \int_{0}^{\frac{3\pi}{2}} cosx d(|cos2x|

    thanks
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: splitting integrals!

    Quote Originally Posted by lawochekel View Post
    i need some help on how to split integrals. for instance this

     \int_{0}^{\frac{3\pi}{2}} cosx d(|cos2x|

    thanks
    If I'm not mistaken you mean you have the differential
    d(|cos(2x)|) = |-2 sin(2x)| dx = 2|sin(2x)| dx

    Where |sin(2x)| = \begin{cases} sin(2x) & 0 \leq x < \frac{\pi}{2} \\ -sin(2x)  & \frac{\pi}{2} \leq x < \pi \end{cases} ...(for 0 \leq x < \pi).

    -Dan
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    Re: splitting integrals!

    pls can you be more explicit cos i don't really get it.
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  4. #4
    Forum Admin topsquark's Avatar
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    Re: splitting integrals!

    Quote Originally Posted by topsquark View Post
    If I'm not mistaken you mean you have the differential
    d(|cos(2x)|) = |-2 sin(2x)| dx = 2|sin(2x)| dx

    Where |sin(2x)| = \begin{cases} sin(2x) & 0 \leq x < \frac{\pi}{2} \\ -sin(2x)  & \frac{\pi}{2} \leq x < \pi \end{cases} ...(for 0 \leq x < \pi).

    -Dan
    You have
    \int_0^{3 \pi/2}cos(x)~d(|cos(2x)|) = \int_0^{3 \pi/2} cos(x) \cdot 2|sin(2x)| dx

    = \int_0^{\pi /2} 2 sin(2x)~cos(x)~dx + \int_{\pi/2}^{\pi}2 cos(x)~(-sin(2x))~dx + \int_{\pi}^{3 \pi/2}{...}

    How do you do the last integral? Is sin(2x) positive or negative?

    -Dan
    Thanks from lawochekel
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    Re: splitting integrals!

    is positive sin(2x)
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