If [x] is greatest integer function, then ∫$$03$$ $$x2$$[x]$$dx=$$

∫$$03$$ $$x2$$[x]$$dx=$$∫$$01$$ $$0dx+$$∫$$12$$ $$x2(1)dx+$$∫$$23$$ $$x2(2)dx=0+x3312+2x3323=83$$−$$13+543$$−$$163=62$$−$$173=15=0015.00$$

Evaluation of definite integrals

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$If [x] is greatest integer function, then \int_{0}^{3} x^{2} [x] d x =$

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Solution

$\begin{array}{l} \int_{0}^{3} x^{2} [x] d x = \int_{0}^{1} 0 d x + \int_{1}^{2} x^{2} (1) d x + \int_{2}^{3} x^{2} (2) d x \\ = 0 + {[\frac{x^{3}}{3}]}_{1}^{2} + 2 {[\frac{x^{3}}{3}]}_{2}^{3} = \frac{8}{3} - \frac{1}{3} + \frac{54}{3} - \frac{16}{3} = \frac{62 - 17}{3} = 15 = 0015.00 \end{array}$

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