${\int }_{-5}^5 |\mathrm{x}+2|\mathrm{dx}$ is equal to

detailed solution

Correct option is D

Here, the given integrand is in the form of absolute function and we define the absolute function as |x|=x,x≥0or |x|=−x,x<0 by using it, we convert the given integrand in simple form and then integrate it.Let, I=∫−55 |x+2|dxIt can be seen that (x+2)≤0 on [−5,−2] and (x+2)≥0 on [-2,5]∴ I=∫−5−2 −(x+2)dx+∫−25 (x+2)dx ∵∫ab f(x)dx=∫ac f(x)dx+∫cb f(x)dx⇒ I=−x22+2x−5−2+x22+2x−25=−(−2)22+2(−2)−(−5)22−2(−5)+(5)22+2(5)−(−2)22−2(−2) =−2−4−252+10+252+10−2+4=−2+4+252−10+252+10−2+4=29