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Question Number 105155 by mohammad17 last updated on 26/Jul/20

Answered by mathmax by abdo last updated on 26/Jul/20

∫_(−∞) ^(+∞)  π e^(−(α^2 /2)) dα =π ∫_(−∞) ^(+∞ )  e^(−(α^2 /2))  dα =_((α/(√2))=x)  π ∫_(−∞) ^(+∞)  e^(−x^2 ) (√2)dx  =π(√2)×(√π)=π(√(2π))

$$\int_{−\infty} ^{+\infty} \:\pi\:\mathrm{e}^{−\frac{\alpha^{\mathrm{2}} }{\mathrm{2}}} \mathrm{d}\alpha\:=\pi\:\int_{−\infty} ^{+\infty\:} \:\mathrm{e}^{−\frac{\alpha^{\mathrm{2}} }{\mathrm{2}}} \:\mathrm{d}\alpha\:=_{\frac{\alpha}{\sqrt{\mathrm{2}}}=\mathrm{x}} \:\pi\:\int_{−\infty} ^{+\infty} \:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \sqrt{\mathrm{2}}\mathrm{dx} \\ $$$$=\pi\sqrt{\mathrm{2}}×\sqrt{\pi}=\pi\sqrt{\mathrm{2}\pi} \\ $$

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