If $[.]$ represents the greatest integer function,

Question:

If $[\cdot]$ 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________

Solution:

$I=\int_{0}^{\sqrt{\pi / 2}}\left(\left[x^{2}\right]+[-\cos x]\right) d x$

$=\int_{0}^{1} 0 d x+\int_{1}^{\sqrt{\pi / 2}} d x+\int_{0}^{\sqrt{\pi / 2}}(-1) d x$

$=\sqrt{\frac{\pi}{2}}-1-\sqrt{\frac{\pi}{2}}=-1$

$\Rightarrow|I|=1$

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