QJEEM23Ma95bc: JEE Main 2023 Mathematics - Class 12 JEEM23Ma95bc Solution: Let denote the greatest integer . Then 2 π ∫ π / 6 5 π / 6 ( 8…

Question

Verified Solution • 2025–26 Session moderate

Let $[t]$ denote the greatest integer $\leq t$ . Then $\frac{2}{π} \int_{π / 6}^{5 π / 6} (8 [cosec x] - 5 [\cot x]) d x$ is equal to __________.

Accepted Answer

\begin{aligned}&amp; \text { Let } \mathrm{I}=\frac{2}{\pi} \int\limits_{\frac{\pi}{6}}^{ \frac{5\pi}{6}}\{8[\operatorname{cosec} x]-5[\cot x]\} d x \\\\&amp; =\frac{2}{\pi}\left[8 \int\limits_{\frac{\pi}{6}}^{\frac{5 \pi}{6}}[\operatorname{cosec} x] d x-5 \int\limits_{\frac{\pi}{6}}^{\frac{5 \pi}{6}}[\cot x] d x\right]\end{aligned}

\left.\begin{array}{r}=\frac{2}{\pi}\left[8 \int\limits_{\pi / 6}^{5 \pi / 6} d x-5\left\{\int\limits_{\pi / 6}^{\pi / 4} d x+\int\limits_{\pi / 4}^{\pi / 2} 0 . d x+\int\limits_{\pi / 2}^{3 \pi / 4}(-1) d x+\right.\right.\left.\left.+\int\limits_{3 \pi / 4}^{5 \pi / 6}(-2) d x\right\}\right]\end{array}\right]

\begin{aligned}&amp; =\frac{2}{\pi}\left[8 \times\left(\frac{5 \pi}{6} - \frac{\pi}{6}\right)-5\left\{\left(\frac{\pi}{4}-\frac{\pi}{6}\right)-\left(\frac{3 \pi}{4}-\frac{\pi}{2}\right)\right\}\right.\left.-2\left(\frac{5 \pi}{6}-\frac{3 \pi}{4}\right)\right] \\\\&amp; =\frac{2}{\pi}\left[\frac{16 \pi}{3}+\frac{5 \pi}{3}\right]=14\end{aligned}

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