Question
The equation $$\sin \,x + x\cos \,x = 0$$ has at least one root in the interval :
A.
$$\left( { - \frac{\pi }{2},\,0} \right)$$
B.
$$\left( {0,\,\pi } \right)$$
C.
$$\left( { - \frac{\pi }{2},\,\frac{\pi }{2}} \right)$$
D.
none of these
Answer :
$$\left( {0,\,\pi } \right)$$
Solution :
$$\eqalign{
& \int {\left( {\sin \,x + x\cos \,x} \right)dx} = - \cos \,x + \int {x\cos \,x\,dx} \cr
& = - \cos \,x + \left\{ {x\sin \,x - \int {\sin \,x\,dx} } \right\} \cr
& = - \cos \,x + x\sin \,x + \cos \,x \cr
& = x\sin \,x \cr} $$
Now, $$x\sin \,x = 0$$ has two roots $$x = 0,\,\pi $$
$$\therefore $$ its derived equation $$\sin \,x + x\cos \,x = 0$$ has a root lying between $$0$$ and $$\pi .$$