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Question

Integrate The function
$$\int_{ - 1}^1 {{x^2} + x - \left[ x \right]dx} $$


Solution

$$\begin{array}{l}\int_{ - 1}^1 {\left[ {{x^2} + x - \left[ x \right]} \right]dx}  = \int_{ - 1}^0 {\left[ {{x^2} + x - \left( { - 1} \right)} \right]dx}  + \int_0^1 {\left[ {{x^2} + x - 0} \right]dx} \\ = \int_{ - 1}^0 {\left[ {{x^2} + x + 1} \right]dx + \int_0^1 {\left[ {{x^2} + x} \right]} dx} \\ = \int_{ - 1}^0 {\left[ {{x^2} + x} \right]dx}  + \int_{ - 1}^0 {1dx + \int_0^\alpha  {0dx}  + \int_\alpha ^2 {1dx} } \\ = \int_{ - 1}^0 { - 1dx}  + 1\left( 1 \right) + 0 + 1\left( {2\alpha } \right)\\ =  - 1\left( 1 \right) + 1 + 2 - \alpha \\ = 2 - \alpha \\ = 2 - \left( {\frac{{\sqrt 3  - 1}}{2}} \right)\\ = \frac{{4 - \sqrt {3 + 1} }}{2}\end{array}$$

Mathematics

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