Let f(x) and g(x) be two functions satisfying f(x^2)+g(4-x)=4x^3 and g(4-x)+g(x)=0.

Question

Let $f(x)$ and $g(x)$ be two functions satisfying $f(x^2)+g(4-x)=4x^3$ and $g(4-x)+g(x)=0$. Then the value of $\displaystyle\int_{-4}^4 f(x^2)\,dx$ is .........

Answer: 512

Accepted Answer

Since $f(x^2)$ is even, $I=\displaystyle\int_{-4}^4f(x^2)\,dx = 2\int_0^4f(x^2)\,dx = 2\int_0^4\big[4x^3-g(4-x)\big]dx = 2^9-2\int_0^4g(4-x)\,dx$

Let $I_1=\displaystyle\int_0^4g(4-x)\,dx$. By $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$: $I_1=\displaystyle\int_0^4g(x)\,dx$ as well, so $2I_1=\displaystyle\int_0^4[g(x)+g(4-x)]dx=0 \implies I_1=0$.

$I = 2^9-0 = 512$

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