Question

EVALUATE$\int\limits^\infty_{-\infty} {e^{-x^{2}}} \, dx$
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Answer 1

I=∫+∞−∞e−x2dx

Then

I2=(∫+∞−∞e−x2dx)(∫+∞−∞e−y2dy)

I2=∫+∞−∞∫+∞−∞e−(x2+y2)dxdy

Now change to polar coordinates

I2=∫+2π0∫+∞0e−r2rdrdθ

The θ integral just gives 2π, while the r integral succumbs to the substitution u=r2

I2=2π∫+∞0e−udu/2=π

So

I=√π

Answer 2

check the attachment