If [.] represents the greatest integer function, then the value of $\left|\int_{0}^{\sqrt{\frac{\pi}{2}}}\left[\left[x^{2}\right]-\cos x\right] d x\right|$ is ____

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Definite Integrals

If [.] represents the greatest integer function, then the value of

\int_{0}^{\frac{π}{2}} [[x^{2}] - cos x] d x

is ____

Answer: 1

For

0 \leq x < 1

one has

[x^{2}] = 0

and

0 < cos x \leq 1

, so

[x^{2}] - cos x

lies in

[- 1, 0)

and its greatest integer is

- 1

. For

1 \leq x \leq \frac{π}{2}

one has

1 \leq x^{2} \leq \frac{π}{2} < 2

, so

[x^{2}] = 1

, while

cos x

lies between

cos π /2

and

cos 1

, both strictly between

0

and

1

; hence

1 - cos x

lies in

(0, 1)

and its greatest integer is

0

. Therefore

\int_{0}^{π /2} [[x^{2}] - cos x] d x = \int_{0}^{1} (- 1) d x + \int_{1}^{π /2} 0 d x = - 1,

and the required value is

1

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