Answer
$$\frac{272}{15}$$
$$or$$
$$18.133$$
Work Step by Step
We are asked to evaluate the integral given using properties of even and odd functions. We are given:
$$\int_{-2}^{2} x^2(x^2+1) dx$$
If we look at a picture of the graph (enclosed below), we will see that the function is an even function. With even functions, we can double the integral while changing the bottom limit from -2 to 0, leaving us with:
$$2\int_{0}^{2} x^2(x^2+1) dx$$
Evaluation:
$2\int_{0}^{2} x^4+x^2 dx$
$2* _{0}^{2} | \frac{1}{5}x^5 + \frac{1}{3}x^3$
$2(\frac{32}{5} + \frac{8}{3})$
$2(\frac{96}{15} + \frac{40}{15})$
$$=\frac{272}{15}$$
