Chapter 12.23, Problem 3MP

Use (5.78e) to show that Then use the result of Problem 2 and Chapter$1, Section $13\text{C}$ to show that ${\displaystyle {\int }_0^1{P}_{2n}}(x)dx=0,n>0,$ and ${\displaystyle {\int }_0^1{P}_{2n+1}}(x)dx=\frac{{(-1)}^n(2n-1)!!}{2^{n+1}(n+1)!}=\left(\begin{array}{c}1/2\\ n+1\end{array}\right)$.