\[\int x(f_{\nu}\widehat{g}_{\nu}+g_{\nu}\widehat{f}_{\nu})\,\mathrm{d}x=\tfrac{%
1}{4}x^{2}(2f_{\nu}\widehat{g}_{\nu}-f_{\nu-1}\widehat{g}_{\nu+1}-f_{\nu+1}%
\widehat{g}_{\nu-1}+2g_{\nu}\widehat{f}_{\nu}-g_{\nu-1}\widehat{f}_{\nu+1}-g_{%
\nu+1}\widehat{f}_{\nu-1}).\]