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

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 $(0, \pi)$.