Question

The value of $12 \int\limits_0^3\left|x^2-3 x+2\right| d x$ is

Answer 1

$ 12 \int\limits_0^3\left| x ^2-3 x +2\right| dx $
$ =12 \int\limits_0^3\left|\left( x -\frac{3}{2}\right)^2-\frac{1}{4}\right| dx$
$ \text { If } x -\frac{3}{2}= t $
$ dx - dt$
$=24 \int\limits_0^{3 / 2}\left| t ^2-\frac{1}{4}\right| dt $
$ =24\left[-\int\limits_0^{1 / 2}\left( t ^2-\frac{1}{4}\right) dt +\int\limits_{1 / 2}^{3 / 2}\left( t ^2-\frac{1}{4}\right) dt \right]=22$