计算

\[ \int_{0}^{\infty}{\frac{1}{\sqrt{x(1+e^{x})}}dx}\]

Solution

\begin{align}
\int_0^\infty\frac{1}{\sqrt{x(1+e^x)}}\mathrm{d}x
&=2\int_0^\infty\frac{1}{\sqrt{1+e^{x^2}}}\mathrm{d}x\\
&=2\int_0^\infty(1+e^{-x^2})^{-1/2}e^{-x^2/2}\;\mathrm{d}x\\
&=2\int_0^\infty\sum_{k=0}^\infty(-\tfrac{1}{4})^k\binom{2k}{k}e^{(2k+1)x^2/2}\;\mathrm{d}x\\
&=\sum_{k=0}^\infty(-\tfrac{1}{4})^k\binom{2k}{k}\sqrt{\frac{2\pi}{2k+1}}
\end{align}

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