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

$(-\pi / 2,0)$ $(0, \pi)$ $(-\pi / 2, \pi / 2)$ none of these

$(0, \pi)$

Consider the function $f(x)$ given by $f(x)=\int(\sin x+x \cos x) d x=x \sin x$ We observe that $f(0)=f(\pi)=0$ Therefore, 0 and $\pi$ are two roots of $f(x)=0$. Consequently, $f^{\prime}(x)=0$ i.e. $\sin x+x \cos x=0$ has at least one root in $(0, \pi)$.