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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