Integrate sqrt(x - x^2)

I've been struggling with this question for hours but has come no closer to solving it. I know it involes the sin inverse function but I cant find a suitable substitution to make that happen.

Would appreciate any help

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- Nov 9th 2009, 07:46 AMadamantineIntegration problem
Integrate sqrt(x - x^2)

I've been struggling with this question for hours but has come no closer to solving it. I know it involes the sin inverse function but I cant find a suitable substitution to make that happen.

Would appreciate any help - Nov 9th 2009, 08:09 AMgalactus
As with most of these, there are various ways to go about it.

$\displaystyle \int\sqrt{x-x^{2}}dx$

$\displaystyle \int\sqrt{x(1-x)}dx$

Make the sub $\displaystyle x=\frac{2u+1}{2}, \;\ dx=du$

and we get $\displaystyle \frac{1}{2}\int\sqrt{1-4u^{2}}du$

Now, there are many routes to take, but a trig sub may be OK.

Can you do it from here?. - Nov 9th 2009, 08:37 AMadamantine
Can I use the trig sub of u = 1/2 sin p?