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If \left [ . \right ] represents the greatest integer function, then the value of \left | \int_{0}^{\sqrt{\frac{\pi }{2}}}\left [ \left [ x^{2} \right ]-\cos x \right ]dx \right | is ____________.
Option: 1 0
Option: 2 1
Option: 3 2
Option: 4 3

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\left|\int_{0}^{\sqrt{\frac{\pi}{2}}}\left[\left[ x ^{2}\right]-\cos x\right] d x \right|

\begin{aligned} I&=\int_{0}^{\sqrt{\pi / 2}}\left(\left[x^{2}\right]+[-\cos x]\right) d x \\ &=\int_{0}^{1} 0 d x+\int_{1}^{\sqrt{\pi / 2}} d x+\int_{0}^{\sqrt{\pi / 2}}(-1) d x \\ &=\sqrt{\frac{\pi}{2}}-1-\sqrt{\frac{\pi}{2}}=-1 \\ &\Rightarrow| I |=1 \end{array}

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Suraj Bhandari

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