integrating $\int \sqrt{2-2\cos(x)} \, dx$ So i am having some trouble getting the solution to the integral:
$\int \sqrt{2-2\cos(x)} \, dx$
i made my first substitution 
$u = 2-2\cos(x)$
$u' = 2\sin(x) \, dx$
then...
$\int \dfrac{1}{\sqrt{4-u}} \, du$
then the next sub of
$s = 4-u$
$s' = -du$
then...
$\int \dfrac{1}{\sqrt{s}} \, ds$
which gave me... 
$2\sqrt{s}$
subbing back $s$...
$2\sqrt{4-u}$
subbing back $u$...
$2\sqrt{2\cos(x)+2}$
but i believe this is incorrect, can someone tell me where i went wrong? 
thanks :)
 A: Hint 
$$2-2\cos x= 2-2\left(\cos^2\left(\frac x 2\right)-\sin^2\left(\frac x 2\right)\right)=4\sin^2\left(\frac x 2\right)$$
A: You have $$\int\sqrt{2-2\cos(x)\,}\,dx.$$ You correctly (though it adds to the work that is necessary) made a substitution with $u=2-2\cos(x)$, so $du=2\sin(x)\,dx$, or equivalently, $du/2\sqrt{1-\cos^2(x)}=dx$. Taking the first equation and solving for $\cos(x)$ gives $1-u/2=\cos(x)$. Now, this gives us $dx=du/\sqrt{1-(1-u+u^2/4)}$, hence we have $$\int\frac{\sqrt{u}}{\sqrt{1-(1-u+u^2/4)}}\,du.$$ Notice the denominator can factor just a bit, so we have $$\int\frac{\sqrt{u}}{\sqrt{u}\sqrt{1-u/4}}\,du.$$ This cancels the $\sqrt{u}$ quite nicely and we get a factor of $2$ outside of the integral after factor $1/4$ out of the denominator. That means we have $$2\int\frac{1}{\sqrt{4-u}}\,du$$ exactly like you have with the exception of the two outside of the integrand. Now you can follow your calculations!
My suggestion (a different technique): multiply by $$\frac{\sqrt{1+\cos(x)}}{\sqrt{1+\cos(x)}}$$ in the integrand (note that this is only $1$, so we're not changing anything). This gives us: $$\sqrt{2}\int\frac{\sin(x)}{\sqrt{1+\cos(x)}}\,dx.$$Now use a $u$-substitution with $u=1+\cos(x)$ and the problem becomes much easier. 
A: here you have not accounted for the $ \sin x $ in terms of $du$.
here we can notice  $ \int\sqrt{2(1-\cos x)} \, dx  $.
now we can write $ 1- \cos x = 2\sin^2{x/2} $.
then your integral becomes
$ 2  \int\sin^2{x/2}\ dx  $.
from which you can proceed further fo the answer.
A: $$
\begin{aligned}
\int \sqrt{2-2 \cos x} d x=&\int \sqrt{2(1-\cos x)} d x \\
&=\int \sqrt{2 \cdot 2 \sin ^2 \frac{x}{2}} d x \\
&=2 \operatorname{sgn}\left(\sin \frac{x}{2}\right) \int \sin \frac{x}{2} d x \\
&=-4 \operatorname{sgn}\left(\sin \frac{x}{2}\right) \cos \frac{x}{2}+C
\end{aligned}
$$
