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  4. If [x] denotes the greatest integer function then ∫ limits03 / 2[x2] d x=

Q. If $[x]$ denotes the greatest integer function then $\int\limits_{0}^{3 / 2}\left[x^{2}\right] d x=$

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Solution:

$\int\limits_{0}^{3 / 2}\left[x^{2}\right] d x=\int\limits_{0}^{1} 0 d x+\int\limits_{1}^{\sqrt{2}} 1 d x+\int\limits_{\sqrt{2}}^{3 / 2} 2 d x$
$=0+(\sqrt{2}-1)+(2(3 / 2)-2 \sqrt{2})=2-\sqrt{2}$.

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