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Evaluate the following problems using properties of integration: $\int\limits_{\large\frac{-\pi}{4}}^{\large\frac{\pi}{4}}x^{3}\cos^{3} xdx$

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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 $\int\limits_{\large\frac{-\pi}{4}}^{\large\frac{\pi}{4}}x^{3}\cos^{3} xdx$
Step 1:
$f(x)=x^{3}\cos^{3} x$
$f(-x)=(-x)^{3}\cos^{3} (-x)$
$\qquad=-x^3 \cos ^3 x$
Step 2:
$\therefore f(x)$ is an odd function
$\int\limits_{\large\frac{-\pi}{4}}^{\large\frac{\pi}{4}}x^{3}\cos^{3} xdx=0$
answered Aug 14, 2013 by