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

The integral of xe^{-(x+y)} is -e^{-y-x}x-e^{-y-x}+C

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integral of xe^{-(x+y)}

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

- e^{- y - x} x - e^{- y - x} + C

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

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

= - e^{- y - x} x - e^{- y - x}

Add a constant to the solution

= - e^{- y - x} x - e^{- y - x} + C

\frac{d}{d x} ((7 - 3 x) (x + 5))

y^{^{'}} + 3 y = cos (x)

4 y^{^{''}} + y^{^{'}} + 8 y = 0

\frac{d}{d x} (x + \frac{1}{x ^{2}} + x^{3})

x \to 5 lim (5 + x^{4})

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