What is the integral of 1000xe^{-0.5x} ?

The integral of 1000xe^{-0.5x} is 4000(-0.5e^{-0.5x}x-e^{-0.5x})+C

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integral of 1000xe^{-0.5x}

\int 1000 x e^{- 0.5 x} d x

4000 (- 0.5 e^{- 0.5 x} x - e^{- 0.5 x}) + C

Solution steps

\int 1000 x e^{- 0.5 x} d x

Take the constant out:

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

= 1000 \cdot \int x e^{- 0.5 x} d x

Apply u-substitution

: \int \frac{2}{0.5} e^{u} u d u

= 1000 \cdot \int \frac{2}{0.5} e^{u} u d u

Take the constant out:

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

= 1000 \cdot \frac{2}{0.5} \cdot \int e^{u} u d u

Simplify

= 1000 \cdot 4 \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 1000 \cdot 4 (e^{u} u - \int e^{u} d u)

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

= 1000 \cdot 4 (e^{u} u - e^{u})

Substitute back

u = - 0.5 x

= 1000 \cdot 4 (e^{- 0.5 x} (- 0.5 x) - e^{- 0.5 x})

Simplify

= 4000 (- 0.5 e^{- 0.5 x} x - e^{- 0.5 x})

Add a constant to the solution

= 4000 (- 0.5 e^{- 0.5 x} x - e^{- 0.5 x}) + C

f (x) = \frac{1}{x - 1}

\frac{d}{d x} (x (3 x - 9)^{3})

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What is the integral of 1000xe^{-0.5x} ?
The integral of 1000xe^{-0.5x} is 4000(-0.5e^{-0.5x}x-e^{-0.5x})+C