Question

By using the properties of definite integrals, evaluate the integral: \(\int_{-5}^{5} |x+2|\,dx\)

Answer 1

Let  \(I\) =\(\int_{-5}^{5} |x+2|\,dx\)

It can be seen that (x+2)≤0 on [−5,−2] and (x+2)≥0 on [−2,5].

∴ \(I=\int^{-2}_{-5}-(x+2)dx+\int^5_{-2}(x+2)dx\)                                                                  \(\bigg(\int^b_af(x)=\int^e_af(x)+\int^b_cf(x)\bigg)\)

I=\(-\bigg[\frac{x^2}{2}+2x\bigg]^{-2}_{-5}+\bigg[\frac{x^2}{2}+2x\bigg]^{5}_{-2}\)

=\(-\bigg[\frac{(-2)^2}{2}+2(-2)-\frac{(-5)^2}{2}-2(-5)\bigg]+-\bigg[\frac{(5)^2}{2}+2(5)-\frac{(-2)^2}{2}-2(-2)\bigg]\)

=-\(\bigg[\)2-4-\(\frac{25}{2}\)+10\(\bigg]\)+\(\bigg[\)\(\frac{25}{2}\)+10-2+4\(\bigg]\)

=-2+4+\(\frac{25}{2}\)-10+\(\frac{25}{2}\)+10-2+4

=29