Mathematics - Limits Question with Solution | TestHub

Let $a,b,c,d$ be four distinct roots of $x^4-4x+3=x\left(x^3-f\prime (1)x^2+f\prime \prime (1)x-4\right)+f(1)$, where $f$ is a monic cubic. Comparing the identity gives $f^{\prime }(1)=f^{\prime \prime }(1)=0$ and $f(1)=3$, hence $f(x)=(x-1{)}^3+3$. If $\int \frac{dx}{\sqrt{\frac{f(x)-3}{x-1}+4}}=\frac{1}{2}g(x)+C$ and $g(3)=\pi /4$, find $g(5)+g(7)$.