Question:
Let $f(x)$ and $g(x)$ be two functions satisfying $f\left(x^{2}\right)$ $+g(4-x)=4 x^{3}$ and $g(4-x)+g(x)=0$, then the
value of $\int_{-4}^{4} f(\mathrm{x})^{2} \mathrm{dx}$ is
Solution:
$\mathrm{I}=2 \int_{0}^{4} f\left(\mathrm{x}^{2}\right) \mathrm{dx} \quad$ Even funtion $\}$
$=2 \int_{0}^{4}\left(4 x^{3}-g(4-x)\right) d x$
$=2\left(\left.\frac{4 x^{4}}{4}\right|_{0} ^{4}-\int_{0}^{4} g(4-x) d x\right)$
$=2(256-0)=512$
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