Evaluate the integral of the quotient of the sine of x and the square root of the quantity 1 plus cosine x, dx.

[tex]\bf \displaystyle \int~\cfrac{sin(x)}{\sqrt{1+cos(x)}}\cdot dx\\\\ -------------------------------\\\\ u=1+cos(x)\implies \cfrac{du}{dx}=-sin(x)\implies \cfrac{du}{-sin(x)}=dx\\\\ -------------------------------\\\\ \displaystyle \int~\cfrac{sin(x)}{\sqrt{u}}\cdot \cfrac{du}{-sin(x)}\implies -\int~\cfrac{1}{\sqrt{u}}\cdot du\implies -\int~u^{-\frac{1}{2}}\cdot du \\\\\\ -\cfrac{u^{\frac{1}{2}}}{\frac{1}{2}}\implies -2u^{\frac{1}{2}}\implies -2\sqrt{1+cos(x)}+C[/tex]