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Homework Help: Integral of square root - Conflicting solutions

  1. Jan 9, 2012 #1
    Can a kind person explain to me why I appear to have two conflicting solutions to:

    [tex] \int^{\frac{1}{\sqrt{2}}}_0dx\sqrt{1-x^2} [/tex]

    Solution 1 : Standard trigonometric substitution: [tex] x=\sin\theta [/tex]

    Integral becomes

    [tex] \int^{\frac{1}{\sqrt{2}}}_0dx\sqrt{1-x^2}=\int^{\sin^{-1}\frac{1}{\sqrt{2}}}_{\sin^{-1}0}\frac{d\theta}{\cos\theta}\sqrt{1-\sin^2\theta}=\int^{\pi/4}_0d\theta=\frac{\pi}{4} [/tex]

    Solution 2 : Integration by parts gives:

    [tex] \int^{\frac{1}{\sqrt{2}}}_0dx\sqrt{1-x^2}=\left.x\sqrt{1-x^2}\right|^{\frac{1}{\sqrt{2}}}_0 + \int^{\frac{1}{\sqrt{2}}}_0dx\frac{x^2}{\sqrt{1-x^2}}=\frac{1}{2}+\int^{\frac{1}{\sqrt{2}}}_0dx \frac{x^2}{\sqrt{1-x^2}} [/tex]

    Adding to the right hand side:

    [tex] 0=\int^{\frac{1}{\sqrt{2}}}_0dx\frac{1}{\sqrt{1-x^2}}-\int^{\frac{1}{\sqrt{2}}}_0dx\frac{1}{\sqrt{1-x^2}} [/tex]

    we get:

    [tex] \int^{\frac{1}{\sqrt{2}}}_0dx\sqrt{1-x^2}=\frac{1}{2} - \int^{\frac{1}{\sqrt{2}}}_0dx\sqrt{1-x^2}+\int^{\frac{1}{\sqrt{2}}}_0dx\frac{1}{\sqrt{1-x^2}} [/tex]

    or rearranging:

    [tex] \int^{\frac{1}{\sqrt{2}}}_0dx\sqrt{1-x^2}=\frac{1}{4}+\frac{1}{2}\int^{\frac{1}{\sqrt{2}}}_0dx \frac{1}{\sqrt{1-x^2}} [/tex]

    Finally since:

    [tex] \sin^{-1} x=\int^x_0\frac{dz}{\sqrt{1-z^2}}\quad |x|\leq1 [/tex]

    we find:

    [tex] \int^{\frac{1}{\sqrt{2}}}_0dx\sqrt{1-x^2}=\frac{1}{4}+\frac{1}{2}\sin^{-1}\frac{1}{\sqrt{2}}=\frac{1}{4}+\frac{\pi}{8} [/tex]

    Any illuminating comments are appreciated.

    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Jan 9, 2012 #2
    Ok forget it, my first integral is completely wrong.
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