Let [.] denote the greatest integer function then the value of _0^1.5 x[x^2],dx is:

Question

Let $[.]$ denote the greatest integer function then the value of $\displaystyle\int_0^{1.5} x[x^2]\,dx$ is:

Accepted Answer

$\int_0^{1.5} x[x^2]\,dx = \int_0^1 x[x^2]\,dx + \int_1^{\sqrt2} x[x^2]\,dx + \int_{\sqrt2}^{1.5} x[x^2]\,dx$

$= \int_0^1 x\cdot0\,dx + \int_1^{\sqrt2} x\cdot1\,dx + \int_{\sqrt2}^{1.5} x\cdot2\,dx$

$= 0 + \left[\dfrac{x^2}{2}\right]_1^{\sqrt2} + [x^2]_{\sqrt2}^{1.5} = \dfrac12(2-1)+(2.25-2) = \dfrac12+\dfrac14 = \dfrac34$

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