$$\begin{align*} I &=\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sin\theta\cos\theta}{\sin^2\theta+\cos^2\theta}d\theta\\&=\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\sin\theta\cos\theta d\theta \\ &=\frac{1}{2}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\sin 2\theta d\theta \\&=\frac{1}{4}\left.\left(-\cos 2\theta\right)\right|_{\frac{\pi}{4}}^{\frac{\pi}{3}}\\&=\boxed{\frac{1}{8}} \end{align*}$$