What is the integral of-x/(e^{-x)} ?

The integral of-x/(e^{-x)} is -e^xx+e^x+C

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integral of-x/(e^{-x)}

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

= - (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

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