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Math Help - Help with Anti-derivative problem

  1. #1
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    Help with Anti-derivative problem

    Find f.


    f ' (x)= 4/{square root(1-2[squared])}, f(1/2)= 1



    Any insight as to how to handle this would be wonderful
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  2. #2
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    Quote Originally Posted by CALsunshine View Post
    Find f.


    f ' (x)= 4/{square root(1-2[squared])}, f(1/2)= 1



    Any insight as to how to handle this would be wonderful
    Is f ' (x) meant to depend on x at all .......?
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  3. #3
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    Hello, CALsunshine!

    I assume there is a typo . . .


    f'(x)\:=\:\frac{4}{\sqrt{1-x^2}},\quad f\left(\frac{1}{2}\right)\:=\: 1

    Find f(x).

    Integrate: . f(x) \;=\;\int\frac{4}{\sqrt{1-x^2}}\,dx \quad\Rightarrow\quad f(x) \;=\;4\sin^{\text{-}1}(x) + C

    Since f\left(\frac{1}{2}\right) \,=\,1, we have: . 1 \;=\;4\sin^{\text{-}1}\left(\frac{1}{2}\right) + C

    . . Then: . 1 \;=\;4\left(\frac{\pi}{6}\right) + C \quad\Rightarrow\quad C \:=\:1 - \frac{2\pi}{3}


    Therefore: . f(x)\;=\;4\sin^{\text{-}1}(x) + 1 - \frac{2\pi}{3}

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