_0^2 [x^2],dx is:

Question

$\displaystyle\int_0^2 [x^2]\,dx$ is:

Accepted Answer

$$\int\limits_0^2 {\left[ {{x^2}} \right]} dx = \int\limits_0^1 {\left[ {{x^2}} \right]dx}  + \int\limits_1^{\sqrt 2 } {\left[ {{x^2}} \right]} dx + $$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $\int\limits_{\sqrt 2 }^{\sqrt 3 } {\left[ {{x^2}} \right]}  + \int\limits_{\sqrt 3 }^2 {\left[ {{x^2}} \right]} dx$$ = \int\limits_0^1 {0dx}  + \int\limits_1^{\sqrt 2 } {1dx}  + \int\limits_{\sqrt 2 }^{\sqrt 3 } {2dx}  + \int\limits_{\sqrt 3 }^2 {3dx} $$ = \left[ x \right]_1^{\sqrt n } + \left[ {2x} \right]_{\sqrt 2 }^{\sqrt 3 } + \left[ {3x} \right]_{\sqrt 3 }^2$$ = \sqrt 2  - 1 + 2\sqrt 3  - 2\sqrt 2  + 6 - 3\sqrt 3 $$ = 5 - \sqrt 3  - \sqrt 2 $

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