What is the integral of (xe^{-x/2})/4 ?

The integral of (xe^{-x/2})/4 is -1/2 e^{-x/2}(x+2)+C

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integral of (xe^{-x/2})/4

\int \frac{x e ^{- \frac{x}{2}}}{4} d x

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

Solution steps

\int \frac{x e ^{- \frac{x}{2}}}{4} d x

Take the constant out:

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

= \frac{1}{4} \cdot \int x e^{- \frac{x}{2}} d x

Apply Integration By Parts

: - 2 e^{- \frac{x}{2}} x - \int - 2 e^{- \frac{x}{2}} d x

= \frac{1}{4} (- 2 e^{- \frac{x}{2}} x - \int - 2 e^{- \frac{x}{2}} d x)

\int - 2 e^{- \frac{x}{2}} d x = 4 e^{- \frac{x}{2}}

= \frac{1}{4} (- 2 e^{- \frac{x}{2}} x - 4 e^{- \frac{x}{2}})

Simplify

\frac{1}{4} (- 2 e^{- \frac{x}{2}} x - 4 e^{- \frac{x}{2}}) : - \frac{1}{2} e^{- \frac{x}{2}} (x + 2)

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

Add a constant to the solution

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

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What is the integral of (xe^{-x/2})/4 ?
The integral of (xe^{-x/2})/4 is -1/2 e^{-x/2}(x+2)+C