Answer to Question #139465 in Calculus for Kembot Lav

Question #139465
integral of e^(-x^2)
1
Expert's answer
2020-10-21T15:18:04-0400

Integral of ex2Integral \ of \ {e^{-x^2}}


Firstly, set the integral to be equal to I

I=ex2dxI = \int{e^{-x^2}dx}


Squaring both sides,

I2=(ex2dx)2=(ex2dx)(ex2dx)I^2 = {(\int{e^{-x^2}dx})}^2 =(\int{e^{-x^2}dx})(\int{e^{-x^2}dx})


Since x is just a dummy variable, we can write,

I2=(ex2dx)(ey2dy)I^2=(\int{e^{-x^2}dx})(\int{e^{-y^2}dy})


Combining, we have,

I2=(e(x2+y2) dx)dyI^2=\int^{\infin}_{-\infin}(\int^{\infin}_{-\infin}{e^{-(x^2+y^2)} \ dx)dy}


I2=e(x2+y2) dxdyI^2=\int^{\infin}_{-\infin}\int^{\infin}_{-\infin}{e^{-(x^2+y^2)} \ dxdy}


Converting from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ)

(r2=x2+y2)(r^2 =x^2 + y^2)

((dxdy=dA=rdrdθ)dxdy =dA = rdrd\,\theta)


I2=er2rdrdθ\therefore I^2 =\iint{e^{-r^2} rdrd\,\theta}


I2=02π(0er2rdr)dθI^2 =\int^{2\pi}_{0}(\int^{\infin}_{0}{e^{-r^2} r\,dr)d\,\theta}


I2=02π(0rer2dr)dθI^2 =\int^{2\pi}_{0}(\int^{\infin}_{0}{re^{-r^2} dr)d\,\theta}


I2=2π(0rer2dr)I^2 =2\pi(\int^{\infin}_{0}{re^{-r^2}dr})


I2=2π0rer2drI^2 =2\pi\int^{\infin}_{0}{re^{-r^2} dr}


Using u substitution,

u=r2u = r^2


dudr=2r\dfrac{du}{dr} = 2r


du=2rdrdu = 2r\,dr



I2=2πu=0reudr\therefore I^2 =2\pi\int^{\infin}_{u =0}{re^{-u}dr}


Multiplying both sides of the equation by 2,

2I2=2πu=02reudr2I^2 =2\pi\int^{\infin}_{u =0}{2re^{-u}dr}


But du = 2rdr

2I2=2πu=0eudu\therefore 2I^2 =2\pi\int^{\infin}_{u =0}{e^{-u}du}


2I2=2π[e0e]2I^2 =2\pi[e^{0}-e^{-\infin}]


2I2=2π(10)2I^2 =2\pi(1-0)


2I2=2π2I^2 =2\pi


I2=πI^2 =\pi


I=πI =\sqrt\pi


 The integral of ex2 is π\therefore \ The \ integral \ of \ e^{−x^2} \ is \ \sqrt{\pi}




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