$\int_{0}^{2} | x^{2} + 2 x - 3 | d x$ is equal to

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Solution

The correct option is A 4
We have
$| x^{2} + 2 x - 3 | = {\begin{matrix} x^{2} + 2 x - 3, & f o r & x < - 3 o r x > 1 - (x^{2} + 2 x - 3), & f o r & - 3 < x < 1 \end{matrix}$
$∴ I = \int_{0}^{2} | x^{2} + 2 x - 3 | d x$
$\Rightarrow I = \int_{0}^{1} | x^{2} + 2 x - 3 | d x + \int_{1}^{2} | x^{2} + 2 x - 3 | d x$
$\Rightarrow I = - \int_{0}^{1} (x^{2} + 2 x - 3) d x + \int_{1}^{2} (x^{2} + 2 x - 3) d x$
$\Rightarrow I = {[\frac{x^{3}}{3} + x^{2} - 3 x]}_{0}^{2} + {[\frac{x^{3}}{3} + x^{2} - 3 x]}_{1}^{2} = 4$

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