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Math Help - How to integrate function?

  1. #1
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    How to integrate function?

    Sorry guys, am very rusty in calculus.
    What method should I use to evaluate this?

    \int{{1\over{\pi\sqrt{x(1-x)}}}}
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  2. #2
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    Quote Originally Posted by chopet View Post
    Sorry guys, am very rusty in calculus.
    What method should I use to evaluate this?

    \int{{1\over{\pi\sqrt{x(1-x)}}}}
    \frac{1}{\sqrt{x(1-x)}}=\frac{1}{\sqrt{ x - x^2}} = \frac{1}{\sqrt{\frac{1}{4} - \left( x - \frac{1}{2} \right)^2}}
    Now let t=x-1/2 the rest is trivial.
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  3. #3
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    Quote Originally Posted by chopet View Post
    Sorry guys, am very rusty in calculus.
    What method should I use to evaluate this?

    \int{{1\over{\pi\sqrt{x(1-x)}}}}
    Define x=\frac1u, the integral becomes to

    - \int {\frac{1}{{u\sqrt {u - 1} }}\,du}.

    Make another substitution defined by \varphi=\sqrt{u-1},

    - \int {\frac{1}<br />
{{u\sqrt {u - 1} }}\,du} = - 2\int {\frac{1}<br />
{{\varphi ^2 + 1}}\,d\varphi } = - 2\arctan \varphi + k.

    Back substitute

    \int {\frac{1}<br />
{{\sqrt {x - x^2 } }}\,dx} = - 2\arctan \frac{{\sqrt {1 - x} }}<br />
{{\sqrt x }} + k.

    ( \pi is just bothering the integrand.)
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