int_{-8}^{8} frac{x^{5}+x^{3}}{4-x^{2}} , dx | vedclass

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Question

(B) Let $I = \int_{-8}^{8} \frac{x^{5}+x^{3}}{4-x^{2}} \, dx$. Define $f(x) = \frac{x^{5}+x^{3}}{4-x^{2}}$. We check if the function is even or odd by

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(B) Let $I = \int_{-8}^{8} \frac{x^{5}+x^{3}}{4-x^{2}} \, dx$. Define $f(x) = \frac{x^{5}+x^{3}}{4-x^{2}}$. We check if the function is even or odd by evaluating $f(-x)$: $f(-x) = \frac{(-x)^{5}+(-x)^{3}}{4-(-x)^{2}} = \frac{-(x^{5}+x^{3})}{4-x^{2}} = -f(x)$. 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,$I = 0$.

$\int_{-8}^{8} \frac{x^{5}+x^{3}}{4-x^{2}} \, dx = $

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