Using the Eigenfunctions of the Previous Problem, the Fourier-Legendre Series $f ( x )$

Using the eigenfunctions of the previous problem, the Fourier-Legendre series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } P _ { n } ( x )$ , where

A) $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } x f ( x ) P _ { n } ( x ) d x$
B) $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x$
C) $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x / 2$
D) $c _ { n } = ( 2 n - 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x / 2$
E) $c _ { n } = ( 2 n - 1 ) \int _ { - 1 } ^ { 1 } x f ( x ) P _ { n } ( x ) d x$