Answer
$$\int_{-8}^{8} \frac{x^{15} d x}{3+\cos ^{2} x}=0$$
Work Step by Step
For the integral $\int_{-8}^{8} \frac{x^{15} d x}{3+\cos ^{2} x}$, we have
$$f(-x)=\frac{(-x)^{15} }{3+\cos ^{2}(-x)}=-\frac{x^{15} d x}{3+\cos ^{2} x}=-f(x).$$
Hence, $f(x)$ is an odd function and then $\int_{-a}^af(x)dx=0$. That is,
$$\int_{-8}^{8} \frac{x^{15} d x}{3+\cos ^{2} x}=0$$
