The value of the integral $\int_{-2}^{2} \frac{\sin ^{2} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}} d x$ (where $[x]$ denotes the greatest integer less than or equal to $x$) is:

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Definite Integrals

The value of the integral

\int_{- 2}^{2} \frac{s i n ^{2} x}{[ \frac{x}{π} ] + \frac{1}{2}} d x

(where

[x]

denotes the greatest integer less than or equal to

x

) is:

Answer: (A)

Over

[- 2, 2]

the value

\frac{x}{π}

stays within

[- 1, 1]

, so

[\frac{x}{π}]

- 1

0

. Pair the integrand at

x

and

- x

: the numerator

sin^{2} x

is even, while

[\frac{- x}{π}] = - [\frac{x}{π}] - 1

turns the denominator

[\frac{x}{π}] + \frac{1}{2}

into its negative. The two contributions are therefore equal and opposite and cancel for every

x

, so the integral is

0

JEE Main · 2019 · 11 Jan 2019 · Shift 1 · Single correct option

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