Answer
$$\frac{2}{3}$$
$$or$$
$$.6666667$$
Work Step by Step
We are asked to evaluate the given integral by using properties of even and odd functions. We are given:
$$\int_{\frac{-π}{2}}^{\frac{π}{2}} sin^2(x)cos(x) dx$$
By looking at a picture of the graph (enclosed below) we will see that the function is even (symmetrical on either side of the y-axis). To find the integral, we need only double it and change the lower limit to zero, leaving us with:
$$2\int_{0}^{\frac{π}{2}} sin^2(x)cos(x) dx$$
Evaluation:
$u=sin(x)$
$du=cos(x)dx$
$2\int_{0}^{\frac{π}{2}} u du$
$2*_{0}^{\frac{π}{2}} | \frac{1}{3} u^3$
$2*_{0}^{\frac{π}{2}} | \frac{1}{3} sin^3(x)$
$2\frac{sin(\frac{π}{2})^3}{3}$
$$=2*\frac{1}{3} = \frac{2}{3} = .6666667$$
