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\displaystyle\begin{aligned} \int_0^{1}e^{-nx^2}dx>\int_0^1 (1-x^2)^n\,dx\Rightarrow \int_0^{1}e^{-nx^2}dx>\int_0^{\frac{\pi}{2}} \cos^{2n+1} t\,dt=\frac{(2n)!!}{(2n+1)!!}
\displaystyle \begin{aligned} \int_0^{\infty}e^{-nx^2}dx<\int_0^{\infty} (x^2+1)^{-n}\,dx\Rightarrow \int_0^{\infty}e^{-nx^2}dx<\int_0^{\frac{\pi}{2}} \cos^{2n-2} t\,dt=\frac{(2n-3)!!}{(2n-2)!!}\frac{\pi}{2}
\displaystyle\begin{aligned} \frac{n}{2n+1}\left[\frac{1}{2n+1} \left(\frac{(2n)!!}{(2n-1)!!} \right)^2\right]<\left(\int_0^{\infty} e^{-x^2}\,dx\right)^2<\frac{\pi^2}{4}\cdot\frac{n}{2n-1}\left[\frac{1}{2n-1} \left(\frac{(2n-2)!!}{(2n-3)!!} \right)^2\right]^{-1}
\displaystyle\begin{aligned} \frac{1}{2}\left(\frac{\pi}{2} \right)\leq\left(\int_0^{\infty} e^{-x^2}\,dx\right)^2\leq\frac{\pi^2}{4}\cdot\frac{1}{2} \left( \frac{2}{\pi}\right)\Rightarrow \left(\int_0^{\infty} e^{-x^2}\,dx\right)^2=\frac{\pi}{4}
= f(x)\cosh (k(x-x)) - 0 + \int_0^x kf(x')\shin (k(x-x')) dx'
= f(x)\cosh (k(x-x)) + \int_0^x kf(x')\shin (k(x-x')) dx' - \int_0^x kf(x')\shin (k(x-x')) dx'
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