Question

Let $f\left(x\right)$ and $g\left(x\right)$ be two functions satisfying $f\left(x^2\right)+g(4-x)=4x^3,g(4-x)+g\left(x\right)=0$, then the value of ${\int }_{-4}^{4}f\left(x^2\right)dx$ is

• A. $512$
• B. $64$
• C. $256$
• D. $0$

Answer 1

Answer: (A)

$512$

${\int }_{-4}^{4}f\left(x^2\right)dx=2{\int }_0^{4}f\left(x^2\right)dx=2{\int }_0^4(4x^3-g(4-x))dx=2\times 256=512$