Let [ t ] denote the greatest integer f . Then 2 6 5 / 6 8 [ c o s e c x ] - 5 [ cot x ] d x is equal to ______ . [2023]

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Solution

Q.
Let $[t]$ denote the greatest integer $\leq f$ . Then $\frac{2}{π} \int_{π / 6}^{5 π / 6} (8 [c o s e c x] - 5 [\cot x]) d x$ is equal to ______ . [2023]

Ans.
(14)

$Let I = \frac{2}{π} \int_{π / 6}^{5 π / 6} (8 [\cos e c x] - 5 [\cot x]) d x$

$= \frac{2}{π} [8 \int_{π / 6}^{5 π / 6} [\cos e c x] d x - 5 \int_{π / 6}^{5 π / 6} [\cot x] d x]$

$= \frac{2}{π} [8 {\int_{π / 6}^{5 π / 6} 1 \cdot d x} - 5 {\int_{π / 6}^{π / 4} 1 \cdot d x + \int_{π / 4}^{π / 2} 0 \cdot d x + \int_{π / 2}^{3 π / 4} (- 1) d x + \int_{3 π / 4}^{5 π / 6} (- 2) d x}]$

$= \frac{2}{π} [8 \times \frac{2 π}{3} - 5 {\frac{π}{12} - \frac{π}{4} - 2 \times \frac{π}{12}}] = \frac{2}{π} [\frac{16 π}{3} + \frac{5 π}{3}] = 14$

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