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Math Help - Trigonometric substitution.

  1. #1
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    Trigonometric substitution.

    hi, i have an example

    \int \sqrt{4^2-4^2\sin ^2\theta .4\cos \theta d\theta }
    it become
    \int 4^2 \cos ^2\theta d\theta
    which later becomes
    4^2\int (\cos 2\theta +1)/2.d\theta

    why is this so?
    Last edited by mr fantastic; May 21st 2011 at 03:04 AM. Reason: Re-titled.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by salohcinseah View Post
    hi, i have an example

    \int \sqrt{4^2-4^2\sin ^2\theta .4\cos \theta d\theta }
    it become
    \int 4^2 \cos ^2\theta d\theta
    which later becomes
    4^2\int (\cos 2\theta +1)/2.d\theta

    why is this so?
    That's only true if the original integral was \int \sqrt{4^2-4^2\sin^2\theta}\cdot 4\cos\theta\,d\theta, which is what I think you wanted to write in the first place.

    First, we see that

    \begin{aligned}\sqrt{4^2-4^2\sin^2\theta}&=\sqrt{4^2(1-\sin^2\theta)}\\&=\sqrt{4^2\cos^2\theta}\\&=4\cos \theta\end{aligned}

    So the integral becomes \int 4\cos\theta\cdot 4\cos\theta\,d\theta = 4^2\int\cos^2\theta\,d\theta

    And by the identity \cos^2\theta=\tfrac{1}{2}(1+\cos(2\theta)), we get 4^2\int\tfrac{1}{2}(1+\cos(2\theta))\,d\theta, which is what we wanted.

    Does this make sense?
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  3. #3
    Junior Member
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    yes, i understand now , the example skip a few step which make me lost
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