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Math Help - Inverse trig

  1. #1
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    Inverse trig

    Differentiate sin^{ - 1}[\frac{(x-5)}{5}] and hence evaulate \int_5^{10} \frac{dx}{\sqrt (10x-x^2)}

    I'm not sure if the number in front of the x in the square root is a 10 or a 30.
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  2. #2
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    Hello, nerdzor!

    Have you done anything yet?


    Differentiate \sin^{\text{-}1}\!\left(\frac{x-5}{5}\right) . and hence evaulate: . \int_5^{10} \frac{dx}{\sqrt{10x-x^2}}
    We have: . y \:=\:\sin^{-1}\left(\frac{x-5}{5}\right)

    Differentiate: . y' \;=\;\frac{\frac{1}{5}}{\sqrt{1-\left(\frac{x-5}{5}\right)^2}} \;=\;\frac{\frac{1}{5}}{\sqrt{1 - \frac{(x-5)^2}{25}}} \;=\;\frac{\frac{1}{5}}{\sqrt{\frac{25-(x-5)^2}{25}}}

    . . y' \;=\;\frac{\frac{1}{5}}{\frac{\sqrt{25-(x-5)^2}}{5}} \;=\;\frac{1}{\sqrt{25 - x^2 + 10x - 25}} \;=\;\frac{1}{\sqrt{10x - x^2}}



    Get it?

    The derivative of \sin^{-1}\left(\frac{x-5}{5}\right) is: . \frac{1}{\sqrt{10x-x^2}}

    So the integral of \frac{1}{\sqrt{10x - x^2}} is: . \sin^{\text{-}1}\left(\frac{x-5}{5}\right) + C


    Hence: . \int^{10}_5\frac{dx}{\sqrt{10x - x^2}}  \;=\;\sin^{\text{-}1}\left(\frac{x-5}{5}\right)\,\bigg]^{10}_5 \;=\;\sin^{\text{-}1}\left(\frac{10-5}{5}\right) - \sin^{\text{-}1}\left(\frac{5-5}{5}\right)

    . . . . . . = \;\; \sin^{\text{-}1}(1) - \sin^{\text{-}1}(0) \;\;=\;\;\frac{\pi}{2} - 0 \;\;=\;\;\frac{\pi}{2}

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  3. #3
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    Quote Originally Posted by nerdzor View Post
    Differentiate sin^{ - 1}[\frac{(x-5)}{5}] and hence evaulate \int_5^{10} \frac{dx}{\sqrt (10x-x^2)}

    I'm not sure if the number in front of the x in the square root is a 10 or a 30.
    It's a 10. I know because here's the answer to the first part. You then use this to find the integral.

    y = \sin^{-1}\Big(\frac{x-5}{5}\Big)

    \Rightarrow \sin y = \frac{x-5}{5}

    \Rightarrow \cos y \frac{dy}{dx} = \frac15

    \Rightarrow \frac{dy}{dx} = \frac{1}{5\cos y}

    = \frac{1}{5\sqrt{1 - \sin^2y}}

    =\frac {1}{\sqrt{25- 25\sin^2y}}

    =\frac{1}{\sqrt{25 - (x-5)^2}}

    =\frac{1}{\sqrt{10x - x^2}}

    Can you complete the second part now?

    Grandad

    PS Beaten by Soroban again - by seconds!
    Last edited by Grandad; May 12th 2009 at 06:04 AM. Reason: Add PS
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