# $\dpi{100} \int_{0}^{\sqrt{2}}\left [ x^{2} \right ]dx\; is$ Option 1) $2-\sqrt{2}\;$ Option 2) $\; \; 2+\sqrt{2}\; \; \; \;$ Option 3) $\sqrt{2}-1\; \;$ Option 4) $\; \sqrt{2}-2\;$

Fundamental Properties of Definite integration -

If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.

$\int_{b}^{a}f\left ( x \right )dx= \int_{a}^{c}f\left ( x \right )dx+\int_{c}^{b}f\left ( x \right )dx$

- wherein

$\int_{0}^{2}[x^{2}]dx$

$\int_{0}^{1}[x^{2}]dx + \int_{0}^{\sqrt{2}}[x^{2}]dx$

=$\int_{0}^{1}0\ dx + \int_{0}^{{2}}1\ dx=(\sqrt{2}-1)$

Option 1)

$2-\sqrt{2}\;$

This is incorrect option

Option 2)

$\; \; 2+\sqrt{2}\; \; \; \;$

This is incorrect option

Option 3)

$\sqrt{2}-1\; \;$

This is correct option

Option 4)

$\; \sqrt{2}-2\;$

This is incorrect option

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