What is the integral of xe^{-y} ?

The integral of xe^{-y} is -xe^{-y}+C

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integral of xe^{-y}

\int x e^{- y} d y

- x e^{- y} + C

Solution steps

\int x e^{- y} d y

Take the constant out:

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

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

Apply u-substitution

: \int - e^{u} d u

= x \cdot \int - e^{u} d u

Take the constant out:

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

= x (- \int e^{u} d u)

Use the common integral:

\int e^{u} d u = e^{u}

= x (- e^{u})

Substitute back

u = - y

= x (- e^{- y})

Simplify

= - x e^{- y}

Add a constant to the solution

= - x e^{- y} + C

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