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Math Help - Integration

  1. #1
    Junior Member
    Joined
    Jan 2009
    Posts
    74

    Integration

    My book did something confusing. I'm not sure how to get from one step to the other.

    Step 1:  y = - \left. \frac{ln|cos(2x)|}{2}\right| _0^\frac{\pi}{6}

    to...

    Step 2:  y = -[-\frac{ln(2)}{2} - 0]

    But I got  cos 2x when  \frac{\pi}{6} is \frac{\pi}{3} and  cos \frac{\pi}{3} = \frac{1}{2}.
    How did it turn into 2 instead of 1/2? And also, where did the  - sign in front of the whole thing come from?

    The original problem is to evaluate:  y = \int_0^ \frac{\pi}{6} tan(2x)dx

    I appreciate the help! Thank you!
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  2. #2
    Senior Member
    Joined
    Dec 2007
    From
    Anchorage, AK
    Posts
    276
    The first (outside) minus sign in y=-[-\frac{\ln(2)}{2}-0] is the minus sign from y=-\left.\frac{\ln|\cos(2x)|}{2}\right|_0^{\frac{\pi}  {6}}; note there is also a minus sign on -\frac{\ln(2)}{2}; this is because \ln\left(\frac{1}{2}\right)=\ln\left(2^{-1}\right)=-\ln{2}, from the basic properties of logarithms (which you should know).

    --Kevin C.
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