What is the integral of e^{-(x^2+y^2)} ?

The integral of e^{-(x^2+y^2)} is e^{-y^2}(sqrt(pi))/2 \erf(x)+C

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integral of e^{-(x^2+y^2)}

\int e^{- (x^{2} + y^{2})} d x

e^{- y^{2}} \frac{π}{2} erf (x) + C

Solution steps

\int e^{- (x^{2} + y^{2})} d x

Simplify

= \int e^{- y^{2} - x^{2}} d x

Simplify

e^{- y^{2} - x^{2}} : e^{- y^{2}} e^{- x^{2}}

= \int e^{- y^{2}} e^{- x^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= e^{- y^{2}} \cdot \int e^{- x^{2}} d x

Use the nonelementary integral:

\int e^{- x^{2}} d x = \frac{π}{2} erf (x)

= e^{- y^{2}} \frac{π}{2} erf (x)

Add a constant to the solution

= e^{- y^{2}} \frac{π}{2} erf (x) + C

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What is the integral of e^{-(x^2+y^2)} ?
The integral of e^{-(x^2+y^2)} is e^{-y^2}(sqrt(pi))/2 \erf(x)+C