What is the integral of xe^{-x}-e^{-x} ?

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

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integral of xe^{-x}-e^{-x}

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

- e^{- x} x + C

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= - e^{- x} x - e^{- x} - (- e^{- x})

Simplify

= - e^{- x} x

Add a constant to the solution

= - e^{- x} x + C

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