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Math Help - Integration using Partial Fractions

  1. #1
    Newbie Mr. Engineer's Avatar
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    Integration using Partial Fractions

    Hey, I need some help with this integral

    \int\frac{1}{\sqrt{-7+8x-x^2}}

    Thanks
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  2. #2
    Member Abu-Khalil's Avatar
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    Observe that \int\frac{dx}{\sqrt{-7+8x-x^2}}=\int\frac{dx}{\sqrt{9-(x-4)^2}}=\frac{1}{3}\int\frac{dx}{\sqrt{1-\left(\frac{x-4}{3}\right)^2}}. So let \sin t=\frac{x-4}{3}\Rightarrow \cos t = \frac{1}{3}, then \frac{1}{3}\int\frac{dx}{\sqrt{1-\left(\frac{x-4}{3}\right)^2}}=\frac{1}{9}\int\frac{\cos t}{\sqrt{1-\sin^2t}}dt=t+K=\frac{1}{9}\arcsin\left(\frac{x-4}{3}\right)+C.
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