# Hard integral( integrate square root of a function)

Please integrate:

$$\int\sqrt{1+2\sqrt{x-x^2 }}dx$$

with respect to $x$.

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The downvotes you are getting are a consequence of the fact that you have showed no effort... –  Nameless Dec 9 '12 at 16:10
And you've used the imperative, like you are giving people orders. –  Simon Hayward Dec 9 '12 at 16:27
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## 2 Answers

Hint:

Notice that $(\sqrt{x}+\sqrt{1-x})^2=x+2\sqrt{x}\sqrt{1-x}+1-x=1+2\sqrt{x-x^2}$

Thus: $\sqrt{1+2\sqrt{x-x^2}}=\sqrt{x}+\sqrt{1-x}$

Using this identity, the integral becomes easier.

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Put $x-x^2=u$, this will transform the integral to

$$-2\int \frac{u}{\sqrt{1-2u}} .$$

Follow it with the transformation $1-2u= z^2,$ the integral falls a part

$$\int (1-z^2) dz .$$

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