Textbook Problem

Proof Suppose that $f\left(a\right)=f\left(b\right)=g\left(a\right)=g\left(b\right)=0$ and the second derivatives of f and g are continuous on the closed interval $\left[a,b\right]$. Prove that

${\displaystyle {\int }_a^{b}f\left(x\right)g^n\left(x\right)dx={\displaystyle \underset{a}{\overset{b}{\int }}{f}^{″}\left(x\right)g\left(x\right)dx.}}$