$$\int_0^{10\pi } {\left| {\sin \,x\,} \right|dx} $$    is-

Solution :
$$I = \int\limits_0^{10\pi } {\left| {\sin \,x\,} \right|dx} = 10\int\limits_0^\pi {\left| {\sin \,x\,} \right|dx} = 10\int\limits_0^\pi {\sin \,x\,dx} $$
[$$\because \left| {\sin \,x} \right|$$   is periodic with period $$\pi $$ and $$\sin \,x > 0$$   if $$0 < x < \pi $$   ]
$$I = 20\int\limits_0^{\frac{\pi }{2}} {\sin \,x\,dx = 20\left[ { - \cos \,x} \right]_0^{\frac{\pi }{2}} = 20} $$