\[\int x(f_{\nu}\widehat{g}_{\nu}-g_{\nu}\widehat{f}_{\nu})\,\mathrm{d}x=\frac{x%
}{2\sqrt{2}}\left(\widehat{f}_{\nu}(f_{\nu+1}+g_{\nu+1})-\widehat{g}_{\nu}(f_{%
\nu+1}-g_{\nu+1})-f_{\nu}(\widehat{f}_{\nu+1}+\widehat{g}_{\nu+1})+g_{\nu}(%
\widehat{f}_{\nu+1}-\widehat{g}_{\nu+1})\right)=\tfrac{1}{2}x(f_{\nu}^{\prime}%
\widehat{f}_{\nu}-f_{\nu}\widehat{f}_{\nu}^{\prime}+g_{\nu}^{\prime}\widehat{g%
}_{\nu}-g_{\nu}\widehat{g}^{\prime}_{\nu}),\]