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Question: \(\int_{0}^{2}{}\lbrack x^{2} - x + 1\rbrack dx\), where [.] denotes greatest integer function...

∫02[x2−x+1]dx\int_{0}^{2}{}\lbrack x^{2} - x + 1\rbrack dx, where [.] denotes greatest integer function

A

7−52\frac{7 - \sqrt{5}}{2}

B

7+52\frac{7 + \sqrt{5}}{2}

C

5−32\frac{\sqrt{5} - 3}{2}

D

None of these

Answer

7−52\frac{7 - \sqrt{5}}{2}

Explanation

Solution

LetI=∫02[x2−x+1]dx=∫01+52[x2−x+1]dx+∫1+522[x2−x+1]dx=∫01+521dx+∫1+5222dx=7−52I = \int_{0}^{2}{}\lbrack x^{2} - x + 1\rbrack dx = \int_{0}^{\frac{1 + \sqrt{5}}{2}}{}\lbrack x^{2} - x + 1\rbrack dx + \int_{\frac{1 + \sqrt{5}}{2}}^{2}{}\lbrack x^{2} - x + 1\rbrack dx = \int_{0}^{\frac{1 + \sqrt{5}}{2}}{}1dx + \int_{\frac{1 + \sqrt{5}}{2}}^{2}{}2dx = \frac{7 - \sqrt{5}}{2}