∫02 x2dx is equal to - Infinity Learn by Sri Chaitanya

A
$5 - \sqrt{2}$
B
$5 - \sqrt{3} - \sqrt{2}$
C
$5 - \sqrt{3}$
D
none of these

Solution:

We have,

$\begin{array}{l} \int_{0}^{2} [x^{2}] d x = \int_{0}^{1} [x^{2}] d x + \int_{1}^{\sqrt{2}} [x^{2}] d x + \int_{\sqrt{2}}^{\sqrt{3}} [x^{2}] d x + \int_{\sqrt{3}}^{2} [x^{2}] d x \\ \Rightarrow \int_{0}^{2} [x^{2}] d x = \int_{0}^{1} 0 d x + \int_{1}^{\sqrt{2}} 1 \cdot d x + \int_{\sqrt{2}}^{\sqrt{3}} 2 d x + \int_{\sqrt{3}}^{2} 3 d x \\ = 0 + 1 (\sqrt{2} - 1) + 2 (\sqrt{3} - \sqrt{2}) + 3 (2 - \sqrt{3}) = 5 - \sqrt{3} - \sqrt{2} \end{array}$

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