Hard integral( integrate square root of a function)

Question

Please integrate:

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

with respect to $x$.

The downvotes you are getting are a consequence of the fact that you have showed no effort...
And you've used the imperative, like you are giving people orders.

Amr · Accepted Answer · 2012-12-27 17:47:57Z

up vote 2 down vote accepted

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.

edited Dec 27 '12 at 17:47

answered Dec 9 '12 at 16:25

Amr
8,82721049

Mhenni Benghorbal · Answer 2 · 2012-12-09 16:19:23Z

up vote 1 down vote

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

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

answered Dec 9 '12 at 16:19

Mhenni Benghorbal
14.2k21026

2 Answers