What is the integral from 0 to 4 of (xe^{-x/2})/4 ?

The integral from 0 to 4 of (xe^{-x/2})/4 is (-12+4e^2)/(4e^2)

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integral from 0 to 4 of (xe^{-x/2})/4

\int_{0}^{4} \frac{x e ^{- \frac{x}{2}}}{4} d x

\frac{- 12 + 4 e ^{2}}{4 e ^{2}}

Decimal Notation

0.59399 \dots

Solution steps

\int_{0}^{4} \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_{0}^{4} 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]_{0}^{4}

= \frac{1}{4} [- 2 e^{- \frac{x}{2}} x - \int - 2 e^{- \frac{x}{2}} d x]_{0}^{4}

\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}}]_{0}^{4}

Compute the boundaries:

- \frac{12}{e ^{2}} + 4

= \frac{1}{4} (- \frac{12}{e ^{2}} + 4)

Simplify

= \frac{- 12 + 4 e ^{2}}{4 e ^{2}}

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What is the integral from 0 to 4 of (xe^{-x/2})/4 ?
The integral from 0 to 4 of (xe^{-x/2})/4 is (-12+4e^2)/(4e^2)