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