I came across this question in my Class Test.
Let $f(x)$ and $g(x)$ be two differentiable functions, defined as
$f(x) = x^2+ xg'(A) + g''(B)$ &
$g(x) = f(A)x^2+ xf'(x) + f''(x)$
Determine the value of the integral $$\int\frac{g(x)}{\sqrt{4x^6-x^4+3x^5}}\,dx.$$
My Attempt: I differentiated $f(x)$ and substituted the value of $f(A)$, $f'(x)$, and $f''(x)$ in g(x). Then differentiated it and put $x=A$ which gives the equation $A(f(A)+2)=0$ which means either $A=0$ or $f(A) =-2$ I made cases and considered $f(A)=-2$ but it made the equations more complicated.
I am stuck here. Is my approach correct? Also suggest if there is any easier method.