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 $\int_{-4}^{4} f(x)^{2} d x$ is
Solution:
$\mathrm{I}=2 \int_{0}^{4} f\left(\mathrm{x}^{2}\right) \mathrm{dx}\{$ Even funtion $\}$
$=2 \int_{0}^{4}\left(4 \mathrm{x}^{3}-\mathrm{g}(4-\mathrm{x})\right) \mathrm{d} \mathrm{x}$
$=2\left(\left.\frac{4 \mathrm{x}^{4}}{4}\right|_{0} ^{4}-\int_{0}^{4} \mathrm{~g}(4-\mathrm{x}) \mathrm{d} \mathrm{x}\right)$
$=2(256-0)=512$
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