Integral of e−x2
Firstly, set the integral to be equal to I
I=∫e−x2dx
Squaring both sides,
I2=(∫e−x2dx)2=(∫e−x2dx)(∫e−x2dx)
Since x is just a dummy variable, we can write,
I2=(∫e−x2dx)(∫e−y2dy)
Combining, we have,
I2=∫−∞∞(∫−∞∞e−(x2+y2) dx)dy
I2=∫−∞∞∫−∞∞e−(x2+y2) dxdy
Converting from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ)
(r2=x2+y2)
(dxdy=dA=rdrdθ)
∴I2=∬e−r2rdrdθ
I2=∫02π(∫0∞e−r2rdr)dθ
I2=∫02π(∫0∞re−r2dr)dθ
I2=2π(∫0∞re−r2dr)
I2=2π∫0∞re−r2dr
Using u substitution,
u=r2
drdu=2r
du=2rdr
∴I2=2π∫u=0∞re−udr
Multiplying both sides of the equation by 2,
2I2=2π∫u=0∞2re−udr
But du = 2rdr
∴2I2=2π∫u=0∞e−udu
2I2=2π[e0−e−∞]
2I2=2π(1−0)
2I2=2π
I2=π
I=π
∴ The integral of e−x2 is π
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