= $$x\log\left [ \sin(x^{2}+1) \right ]^{\frac{1}{2}}$$
Using property of log and then product rule we get,
$$\frac{dy}{dx} = \frac{1}{2}\left [ x\left ( \frac{1}{\sin(x^{2}+1)}\cos(x^{2}+1)2x \right )+\log(\sin(x^{2}+1)) \right ]$$ $$dy/dx = x^{2}\cot(x^{2}+1)+\frac{\log(\sin(x^{2}+1))}{2}$$ This was my answer. However the answer given is $$(x^{2}\cot(x^{2}+1))+ \log \sqrt{\sin(x^{2}+1)}$$
They are the same: $$\frac{\log a}{2} = \frac12 \log a = \log(a^{1/2}) = \log(\sqrt a)$$