JEE MAIN - Mathematics (2002 - No. 19)
$$\int\limits_0^{10\pi } {\left| {\sin x} \right|dx} $$ is
$$20$$
$$8$$
$$10$$
$$18$$
Explanation
$$I = \int\limits_0^{10\pi } {\left| {\sin x} \right|} dx$$
$$ = 10\int\limits_0^\pi {\left| {\sin x\,} \right|} \,dx$$
$$ = 10\int\limits_0^\pi {\sin \,x\,dx} $$
$$\left[ \, \right.$$ as $$\left| {\sin x} \right|$$ is periodic with period $$\pi $$
and $$sin$$ $$x > 0$$ if $$0 < x$$ $$ < \pi $$ $$\left. \, \right]$$
$$I = 20\int\limits_0^{\pi /2} {\sin x} \,dx$$
$$ = 20\left[ { - \cos \,x_0^{\pi /2}} \right] = 20$$
$$ = 10\int\limits_0^\pi {\left| {\sin x\,} \right|} \,dx$$
$$ = 10\int\limits_0^\pi {\sin \,x\,dx} $$
$$\left[ \, \right.$$ as $$\left| {\sin x} \right|$$ is periodic with period $$\pi $$
and $$sin$$ $$x > 0$$ if $$0 < x$$ $$ < \pi $$ $$\left. \, \right]$$
$$I = 20\int\limits_0^{\pi /2} {\sin x} \,dx$$
$$ = 20\left[ { - \cos \,x_0^{\pi /2}} \right] = 20$$
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