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STD 12 - 7.2 definite integral — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 7.2 definite integral4 Marks
Question
Let $[t]$ denote the greatest integer function. If $\int \limits_0^{2.4}\left[x^2\right] d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$, then $\alpha+\beta+\gamma+$ $\delta$ is equal to $..............$.
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Answer

a
$\int \limits_0^1 0 dx +\int \limits_1^{\sqrt{2}} 1 dx +\int \limits_{\sqrt{2}}^{\sqrt{3}} 2 dv +\int \limits_{\sqrt{3}}^2 3 dx +\int \limits_2^{\sqrt{5}} 4 dx +\int \limits_{\sqrt{5}}^{2 \cdot 4} 5 dx$

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

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

$\alpha+\beta+\gamma+\delta=9-1-1-1=6$

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