Area by integration

Let $f(x) = |x+1|+|x+2|-|2x+3|$ . Then we have:

$\begin{cases} \text{For } -2 \le x < - \frac 32: & f(x) = - x - 1 + x + 2 + 2x + 3 = 2x+4 \\ \text{For } - \frac 32\le x < - 1: & f(x) = - x - 1 + x + 2 - 2x - 3 = -2x-2 \\ \text{For } x \ge -1: & f(x) = x+1 + x+2 - 2x -3 = 0 \end{cases}$

Therefore,

$\begin{aligned} \int_{-2}^5 f(x)\ dx & = \int_{-2}^{-\frac 32} f(x)\ dx + \int_{-\frac 32}^{-1} f(x)\ dx + \int_{-1}^5 f(x)\ dx \\ & = \int_{-2}^{-\frac 32} 2x+4\ dx + \int_{-\frac 32}^{-1} (-2x-2)\ dx + \int_{-1}^5 0\ dx \\ & = x^2 + 4x \bigg|_{-2}^{-\frac 32} - x^2 - 2x \bigg|_{-\frac 32}^{-1} + 0 \\ & = \frac 94 - 6 - 4 + 8 - 1 + 2 + \frac 94 - 3 \\ & = \frac 12 = \boxed {0.5} \end{aligned}$