# Thread: How to integrate function?

1. ## How to integrate function?

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

$\int{{1\over{\pi\sqrt{x(1-x)}}}}$

2. Originally Posted by chopet
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.

3. Originally Posted by chopet
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 $\varphi=\sqrt{u-1},$

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

Back substitute

$\int {\frac{1}
{{\sqrt {x - x^2 } }}\,dx} = - 2\arctan \frac{{\sqrt {1 - x} }}
{{\sqrt x }} + k.$

( $\pi$ is just bothering the integrand.)