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

The integral of x*e^{-sx} is 1/(s^2)(-se^{-sx}x-e^{-sx})+C

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integral of x*e^{-sx}

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = - s x

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial y} (\frac{e ^{y}}{e ^{x} + e ^{y}})

\frac{d}{d x} (3 x^{2} - 2 x + 5)

\int (x y e^{x z}) d z

\int x I n x d x

\int - \frac{1}{( x - 1 ) ^{2}} d x

What is the integral of x*e^{-sx} ?
The integral of x*e^{-sx} is 1/(s^2)(-se^{-sx}x-e^{-sx})+C