Evaluate the integral: int_{-4 pi}^{4 pi} tan | vedclass

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(D) Let $f(x) = 	an ^9 x \sin ^6 x \cos ^3 x$. We check if the function is even or odd by evaluating $f(-x)$: $f(-x) = [	an(-x)]^9 [\sin(-x)]^6 [\co

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(D) Let $f(x) = 	an ^9 x \sin ^6 x \cos ^3 x$. We check if the function is even or odd by evaluating $f(-x)$: $f(-x) = [	an(-x)]^9 [\sin(-x)]^6 [\cos(-x)]^3$ Since $	an(-x) = -	an x$,$\sin(-x) = -\sin x$,and $\cos(-x) = \cos x$,we have: $f(-x) = (-	an x)^9 (-\sin x)^6 (\cos x)^3$ $f(-x) = -	an^9 x \cdot \sin^6 x \cdot \cos^3 x = -f(x)$. Since $f(-x) = -f(x)$,the function $f(x)$ is an odd function. According to the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^{a} f(x) \, dx = 0$. Therefore,$\int_{-4 \pi}^{4 \pi} 	an ^9 x \sin ^6 x \cos ^3 x \, dx = 0$.

Evaluate the integral: $\int_{-4 \pi}^{4 \pi} \tan ^9 x \sin ^6 x \cos ^3 x \, dx$

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