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NIMCET 2015 Mathematics PYQ — , where denotes the greatest integer function, is equal to:… | Mathem Solvex

A.
$\frac{\pi}{2}$
B.
$1$
C.
$-1$
D.
$-\frac{\pi}{2}$
(Correct Answer)

Correct Answer:

$-\frac{\pi}{2}$

Explanation

I = \int_{0}^{\pi} [\cot x] \, dx \quad \dots(1)

Using $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$ :

I = \int_{0}^{\pi} [\cot(\pi - x)] \, dx

I = \int_{0}^{\pi} [-\cot x] \, dx \quad \dots(2)

Adding $(1)$ and $(2)$ :

2I = \int_{0}^{\pi} ([\cot x] + [-\cot x]) \, dx

Using property: $[x] + [-x] = -1$ (for $x \notin \mathbb{Z}$ ):

2I = \int_{0}^{\pi} (-1) \, dx

2I = [-x]_{0}^{\pi}

2I = -\pi

I = -\frac{\pi}{2}

Correct Option: (d)

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NIMCET 2015 — Mathematics PYQ

Choose the correct answer:

Explanation

Explanation