# Evaluate the following problems using properties of integration: $\int\limits_{\large\frac{-\pi}{4}}^{\large\frac{\pi}{4}}x \sin^{2}xdx$

Toolbox:
• $\int \limits_{-a}^a f(x) dx=2 \int \limits_0^a f(x) dx$ if f is an even function
• $\int \limits_{-a}^a f(x) dx=0$ if f is an odd function
Given $\int\limits_{\large\frac{-\pi}{4}}^{\large\frac{\pi}{4}}x \sin^{2}xdx$
Step 1:
$f(x)=x \sin ^2 x$
$f(-x)=(-x) \sin ^2 (-x)$
$\qquad=(-x)(-\sin x)^2$
$\qquad=-x \sin ^2 x$
Step 2:
$\therefore f(x)$ is an odd function
$\int\limits_{\large\frac{-\pi}{4}}^{\large\frac{\pi}{4}}x \sin^{2}xdx=0$
answered Aug 14, 2013 by
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