Question

If $kϵN$ and $I_k={\int }_{-2k\pi }^{2k\pi }|\sin x|\left[\sin x\right]dx$, (where [.] denotes greatest integer function), then

• A. $I_{20}=-80$
• B. $I_{20}=-60$
• C. ${\sum }_{k=1}^{100}{I}_k=-20200$
• D. ${\sum }_{k=1}^{100}{I}_k=-10100$

Answer 1

Answer: (A), (C)

${\sum }_{k=1}^{100}{I}_k=-20200$

$I_k={\int }_{-2k\pi }^{2k\pi }|\sin x|\left[\sin x\right]dx{I}_k={\int }_{-2k\pi }^{2k\pi }|\sin x|[-\sin x]dx$
On adding the above equations we get-
$2I_k={\int }_{-2k\pi }^{2k\pi }|\sin x|\left(\right[\sin x]+[-\sin x\left]\right)dx2I_k=2{\int }_0^{2k\pi }|\sin x|(-1)dx{I}_k=-2k{\int }_0^{\pi }|\sin x|dx{I}_k=-4k$