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JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Definite Integrals

Let

[t]

denotes 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

Answer: 14

Solution

On

(\frac{π}{6}, \frac{5 π}{6})

,

sin x \in (\frac{1}{2}, 1]

, so

cosec x \in [1, 2)

and

[cosec x] = 1

throughout. Hence

\int_{π /6}^{5 π /6} 8 [cosec x] d x = 8 (\frac{5 π}{6} - \frac{π}{6}) = \frac{16 π}{3} .

For

[cot x]

,

cot x

decreases from

3

to

- 3

, giving

[cot x] = 1

on

(\frac{π}{6}, \frac{π}{4})

,

0

on

(\frac{π}{4}, \frac{π}{2})

,

- 1

on

(\frac{π}{2}, \frac{3 π}{4})

and

- 2

on

(\frac{3 π}{4}, \frac{5 π}{6})

, with lengths

\frac{π}{12}, \frac{π}{4}, \frac{π}{4}, \frac{π}{12}

. Thus

\int_{π /6}^{5 π /6} [cot x] d x = \frac{π}{12} - \frac{π}{4} - \frac{2 π}{12} = - \frac{π}{3} .

Therefore

\frac{2}{π} \int_{π /6}^{5 π /6} (8 [cosec x] - 5 [cot x]) d x = \frac{2}{π} (\frac{16 π}{3} + \frac{5 π}{3}) = \frac{2}{π} \cdot \frac{21 π}{3} = 14.

JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Integer answer

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