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

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

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integral of-x*e^{-x}

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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