Here is the graph of this function : $\int_{-4}^{4}|x-3|$
Observing here we find that at $x=3$ function is $0$ and before that it is negative therefore we can work out integration of this function by removing absolute function as follows :
$\Rightarrow $ $\int_{-4}^{3}-(x-3)+\int_{3}^{4}(x-3)$
$\Rightarrow $ $\int_{-4}^{3}(3-x)\int_{3}^{4}(x-3)$
$\Rightarrow $ $[3x-\frac{x^2}{2}]_{-4}^{3}+[\frac{x^2}{2}-3x]_{3}^4$
$\Rightarrow \ 25$