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\begin{align}
\sum_{\ell = 0}^{\infty}h^{\ell}\int_{-1}^{0}{\rm P}_{\ell}\pars{x}
&=\int_{-1}^{0}{\dd x \over \root{1 - 2xh + h^{2}}}
=\left.{\root{1 - 2xh + h^{2}} \over -h}\right\vert_{x\ =\ 0}^{x\ =\ 1}
={1 \over h}\bracks{\root{1 + h^{2}} - 1 + h}
\\[3mm]&={1 \over h}\bracks{\sum_{\ell = 0}^{\infty}
{1/2 \choose \ell}h^{2\ell} - 1 + h}=
1 + \sum_{\ell = 1}^{\infty}
{1/2 \choose \ell}h^{2\ell - 1}
\end{align}
\begin{align}
{1/2 \choose \ell} &= {\Gamma\pars{3/2} \over \ell!\,\Gamma\pars{3/2 - \ell}}
= {\root{\pi} \over 2}\,{1 \over \ell!\,\Gamma\pars{3/2 - \ell}}
\\[3mm]&= {\root{\pi} \over 2}\,
{1 \over \ell!\braces{\pi/\bracks{\Gamma\pars{\ell - 1/2}}
\sin\pars{\pi\bracks{\ell - 1/2}}}}
= {1 \over 2\root{\pi}}\,
{\pars{-1}^{\ell + 1}\Gamma\pars{\ell - 1/2} \over \ell!}
\\[2mm]&= {1 \over \root{\pi}}\,
{\pars{-1}^{\ell + 1}\Gamma\pars{\ell + 1/2} \over \pars{2\ell - 1}\ell!}
= {1 \over \root{\pi}}\,
{\pars{-1}^{\ell + 1} \over \pars{2\ell - 1}\ell!}\,
{\Gamma\pars{2\ell}\root{\pi} \over 2^{2\ell - 1}\Gamma\pars{\ell}}
\\[3mm]&=
{\pars{-1}^{\ell + 1} \over \pars{2\ell - 1}\ell!}\,
{\pars{2\ell - 1}! \over 2^{2\ell - 1}\pars{\ell - 1}!}
\end{align}
$$\color{#00f}{\large\left\lbrace%
\begin{array}{rcl}
\int_{-1}^{0}{\rm P}_{0}\pars{x}\,\dd x & = & 1
\\[2mm]
\int_{-1}^{0}{\rm P}_{2\ell - 1}\pars{x}\,\dd x & = & \pars{-1}^{\ell + 1}\,
{2^{-\pars{2\ell - 1}} \over 2\ell - 1}\,{2\ell - 1 \choose \ell}\,,\quad
\ell = 1,2,3,\ldots
\\[2mm] && 0\ \mbox{otherwise}
\end{array}\right.}
$$