What is the integral of ye^{-(x^2+y^2)}(1-2x^2) ?

The integral of ye^{-(x^2+y^2)}(1-2x^2) is -1/2 e^{-x^2-y^2}+x^2e^{-x^2-y^2}+C

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integral of ye^{-(x^2+y^2)}(1-2x^2)

\int y e^{- (x^{2} + y^{2})} (1 - 2 x^{2})

- \frac{1}{2} e^{- x^{2} - y^{2}} + x^{2} e^{- x^{2} - y^{2}} + C

Solution steps

\int y e^{- (x^{2} + y^{2})} (1 - 2 x^{2}) d y

Take the constant out:

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

= (1 - 2 x^{2}) \cdot \int e^{- (x^{2} + y^{2})} y d y

\int e^{- (x^{2} + y^{2})} y d y = - \frac{1}{2} e^{- x^{2} - y^{2}}

= (1 - 2 x^{2}) (- \frac{1}{2} e^{- x^{2} - y^{2}})

Simplify

(1 - 2 x^{2}) (- \frac{1}{2} e^{- x^{2} - y^{2}}) : - \frac{1}{2} e^{- x^{2} - y^{2}} + x^{2} e^{- x^{2} - y^{2}}

= - \frac{1}{2} e^{- x^{2} - y^{2}} + x^{2} e^{- x^{2} - y^{2}}

Add a constant to the solution

= - \frac{1}{2} e^{- x^{2} - y^{2}} + x^{2} e^{- x^{2} - y^{2}} + C

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What is the integral of ye^{-(x^2+y^2)}(1-2x^2) ?
The integral of ye^{-(x^2+y^2)}(1-2x^2) is -1/2 e^{-x^2-y^2}+x^2e^{-x^2-y^2}+C