The equation $\sin x+x\cos x=0$   has at least one root in the interval 

 Consider the function $f\left(x\right)$ given by 

$f\left(x\right)=\int (\sin x+x\cos x)dx=x\sin x$

We observe that  

$f\left(0\right)=f\left(\pi \right)=0$ 

Therefore, 0 and $\pi$ are two roots of $f\left(x\right)=0$ 

Consequently $f^{\mathrm{\prime }}\left(x\right)=0\text{ i.e. }\sin x+x\cos x=0$  has at least one 

root in $(0,\pi )\text{. }$