What is the integral of 5xe^{-5x} ?

The integral of 5xe^{-5x} is -e^{-5x}x-1/5 e^{-5x}+C

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integral of 5xe^{-5x}

\int 5 x e^{- 5 x} d x

- e^{- 5 x} x - \frac{1}{5} e^{- 5 x} + C

Solution steps

\int 5 x e^{- 5 x} d x

Take the constant out:

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

= 5 \cdot \int x e^{- 5 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{25} d u

= 5 \cdot \int \frac{e ^{u} u}{25} d u

Take the constant out:

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

= 5 \cdot \frac{1}{25} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 5 \cdot \frac{1}{25} (e^{u} u - \int e^{u} d u)

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

= 5 \cdot \frac{1}{25} (e^{u} u - e^{u})

Substitute back

u = - 5 x

= 5 \cdot \frac{1}{25} (e^{- 5 x} (- 5 x) - e^{- 5 x})

Simplify

5 \cdot \frac{1}{25} (e^{- 5 x} (- 5 x) - e^{- 5 x}) : - e^{- 5 x} x - \frac{1}{5} e^{- 5 x}

= - e^{- 5 x} x - \frac{1}{5} e^{- 5 x}

Add a constant to the solution

= - e^{- 5 x} x - \frac{1}{5} e^{- 5 x} + C

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What is the integral of 5xe^{-5x} ?
The integral of 5xe^{-5x} is -e^{-5x}x-1/5 e^{-5x}+C