QJEEM21Ma80d1: JEE Main 2021 Mathematics - Class 12 JEEM21Ma80d1 Solution: The value of the integral, ∫ 1 3 [ x 2 − 2 x − 2 ] d x , where…

Question

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The value of the integral,

\int_{1}^{3} [x^{2} - 2 x - 2] d x

, where [x] denotes the greatest integer less than or equal to x, is :

Accepted Answer

I = \int_{1}^{3} - 3 d x + \int_{1}^{3} [{(x - 1)}^{2}] d x

Put x

-

1 = t ; dx = dt

I = (- 6) + \int_{0}^{2} [t^{2}] d t

I = - 6 + \int_{0}^{1} 0 d t + \int_{1}^{\sqrt{2}} 1 d t + \int_{\sqrt{2}}^{\sqrt{3}} 2 d t + \int_{\sqrt{3}}^{2} 3 d t

I = - 6 + (\sqrt{2} - 1) + 2 \sqrt{3} - 2 \sqrt{2} + 6 - 3 \sqrt{3}

I = - 1 - \sqrt{2} - \sqrt{3}

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