Correct Answer - Option 3 : 29
Concept:
The Modulus Function '| |' is defined as: \(\rm |x|=\left\{\begin{matrix}\rm \ \ \ x, &\rm x \geq 0\\ \rm -x, &\rm x<0\\\end{matrix}\right.\).
Calculation:
Consider |x + 2|.
|x + 2| = x + 2 for x ≥ -2.
|x + 2| = - x - 2 for x < -2.
Now, \(\rm \int_{ - 5}^{\ \ 5} \left| {x + 2} \right|dx\):
= \(\rm \int_{ - 5}^{-2} (-x-2)\ dx+\int_{-2}^{\ \ 5} (x+2)\ dx\)
= \(\rm \left[\frac{-x^2}{2}-2x\right]_{ - 5}^{-2}+\left[\frac{x^2}{2}+2x\right]_{-2}^{\ \ \ 5}\)
= \(\rm \left(\frac{-4}{2}+4\right)-\left(\frac{-25}{2}+10\right)+\left(\frac{25}{2}+10\right)-\left(\frac{4}{2}-4\right)\)
= 29.
