The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution $t=\tan(x/2)$ will convert any rational function of $\sin(x)$ and $\cos(x)$ into an ordinary rational function of $t$. If $t=\tan(x/2),\; -\pi < x < \pi,$ then it can be shown that $$\cos\left(\frac{x}{2}\right)=\frac{1}{\sqrt{1+t^2}},\qquad \sin\left(\frac{x}{2}\right)=\frac{t}{\sqrt{1+t^2}},$$ $$\cos(x)=\frac{1-t^2}{1+t^2},\qquad \sin(x)=\frac{2t}{1+t^2},$$ and $$dx=\frac{2}{1+t^2}dt.$$

Use the substitution given above to transform the following integral into a rational function of $t$ and then evaluate the integral: $$\int_{\pi/3}^{\,\pi/2} {\frac{3}{1+\sin(x)-\cos(x)}}\, dx$$