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Question Number 55364 by peter frank last updated on 22/Feb/19

$$\mathrm{prove}\mathrm{that}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=1$$$$\mathrm{such}\mathrm{that}\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\sqrt{\mathrm{n}}\beta (\frac{\mathrm{n}}{2},\frac{1}{2})}{(1+\frac{{\mathrm{x}}^2}{\mathrm{n}})}^{-\frac{1}{2}(1+\mathrm{n})}$$$$\mathrm{and}\beta (\frac{\mathrm{n}}{2},\frac{1}{2})={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{x}}^{\frac{\mathrm{n}}{2}-1}}{{(1+\mathrm{x})}^{\frac{3}{2}}}\mathrm{dx}$$

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