Let [ t ] denote the greatest integer function. If 0 2 . 4 [ x 2 ] ? d x = 2 + 3 + 5 , then is equal to _________ . [2023]

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Solution

Q.
Let $[t]$ denote the greatest integer function. If $\int_{0}^{2.4} [x^{2}] d x = α + β \sqrt{2} + γ \sqrt{3} + δ \sqrt{5},$ then $α + β + γ + δ$ is equal to _________ . [2023]

Ans.
(6)

We have,

$\int_{0}^{2.4} [x^{2}] d x = \int_{0}^{1} 0 d x + \int_{1}^{\sqrt{2}} 1 d x + \int_{\sqrt{2}}^{\sqrt{3}} 2 d x + \int_{\sqrt{3}}^{2} 3 d x + \int_{2}^{\sqrt{5}} 4 d x + \int_{\sqrt{5}}^{2.4} 5 d x$

$= \sqrt{2} - 1 + 2 (\sqrt{3} - \sqrt{2}) + 3 (2 - \sqrt{3}) + 4 (\sqrt{5} - 2) + 5 (2.4 - \sqrt{5})$

$= 9 - \sqrt{2} - \sqrt{3} - \sqrt{5}$

$∴$ $α + β + γ + δ = 9 - 1 - 1 - 1 = 6$

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