What is the integral from 0 to y of 2xye^{-x-y} ?

The integral from 0 to y of 2xye^{-x-y} is 2y(-ye^{-2y}-e^{-2y}+e^{-y})

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integral from 0 to y of 2xye^{-x-y}

\int_{0}^{y} 2 x y e^{- x - y} d x

2 y (- y e^{- 2 y} - e^{- 2 y} + e^{- y})

Solution steps

\int_{0}^{y} 2 x y e^{- x - y} d x

Take the constant out:

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

= 2 y \cdot \int_{0}^{y} e^{- y - x} x d x

Apply Integration By Parts

: [- e^{- y - x} x - \int - e^{- y - x} d x]_{0}^{y}

= 2 y [- e^{- y - x} x - \int - e^{- y - x} d x]_{0}^{y}

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

= 2 y [- e^{- y - x} x - e^{- y - x}]_{0}^{y}

Compute the boundaries:

- y e^{- 2 y} - e^{- 2 y} + e^{- y}

= 2 y (- y e^{- 2 y} - e^{- 2 y} + e^{- y})

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What is the integral from 0 to y of 2xye^{-x-y} ?
The integral from 0 to y of 2xye^{-x-y} is 2y(-ye^{-2y}-e^{-2y}+e^{-y})