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

The integral of xe^{-cx} is 1/(c^2)(-ce^{-cx}x-e^{-cx})+C

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integral of xe^{-cx}

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

\frac{1}{c ^{2}} (- c e^{- c x} x - e^{- c x}) + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{u} u}{c ^{2}} d u

= \int \frac{e ^{u} u}{c ^{2}} d u

Take the constant out:

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

= \frac{1}{c ^{2}} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= \frac{1}{c ^{2}} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{c ^{2}} (e^{u} u - e^{u})

Substitute back

u = - c x

= \frac{1}{c ^{2}} (e^{- c x} (- c x) - e^{- c x})

Simplify

= \frac{1}{c ^{2}} (- c e^{- c x} x - e^{- c x})

Add a constant to the solution

= \frac{1}{c ^{2}} (- c e^{- c x} x - e^{- c x}) + C

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What is the integral of xe^{-cx} ?
The integral of xe^{-cx} is 1/(c^2)(-ce^{-cx}x-e^{-cx})+C