Menu Close

Question-105155




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

Leave a Reply

Your email address will not be published. Required fields are marked *