What is the integral from 0 to 20 of 1/16 xe^{-x/4} ?

The integral from 0 to 20 of 1/16 xe^{-x/4} is (-96+16e^5)/(16e^5)

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integral from 0 to 20 of 1/16 xe^{-x/4}

\int_{0}^{20} \frac{1}{16} x e^{- \frac{x}{4}} d x

\frac{- 96 + 16 e ^{5}}{16 e ^{5}}

Decimal

0.95957 \dots

Solution steps

\int_{0}^{20} \frac{1}{16} x e^{- \frac{x}{4}} d x

Take the constant out:

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

= \frac{1}{16} \cdot \int_{0}^{20} x e^{- \frac{x}{4}} d x

Apply Integration By Parts

: [- 4 e^{- \frac{x}{4}} x - \int - 4 e^{- \frac{x}{4}} d x]_{0}^{20}

= \frac{1}{16} [- 4 e^{- \frac{x}{4}} x - \int - 4 e^{- \frac{x}{4}} d x]_{0}^{20}

\int - 4 e^{- \frac{x}{4}} d x = 16 e^{- \frac{x}{4}}

= \frac{1}{16} [- 4 e^{- \frac{x}{4}} x - 16 e^{- \frac{x}{4}}]_{0}^{20}

Compute the boundaries:

- \frac{96}{e ^{5}} + 16

= \frac{1}{16} (- \frac{96}{e ^{5}} + 16)

Simplify

= \frac{- 96 + 16 e ^{5}}{16 e ^{5}}

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What is the integral from 0 to 20 of 1/16 xe^{-x/4} ?
The integral from 0 to 20 of 1/16 xe^{-x/4} is (-96+16e^5)/(16e^5)