What is the integral of 8xe^{-8x} ?

The integral of 8xe^{-8x} is -e^{-8x}x-1/8 e^{-8x}+C

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integral of 8xe^{-8x}

\int 8 x e^{- 8 x} d x

- e^{- 8 x} x - \frac{1}{8} e^{- 8 x} + C

Solution steps

\int 8 x e^{- 8 x} d x

Take the constant out:

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

= 8 \cdot \int x e^{- 8 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{64} d u

= 8 \cdot \int \frac{e ^{u} u}{64} d u

Take the constant out:

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

= 8 \cdot \frac{1}{64} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 8 \cdot \frac{1}{64} (e^{u} u - \int e^{u} d u)

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

= 8 \cdot \frac{1}{64} (e^{u} u - e^{u})

Substitute back

u = - 8 x

= 8 \cdot \frac{1}{64} (e^{- 8 x} (- 8 x) - e^{- 8 x})

Simplify

8 \cdot \frac{1}{64} (e^{- 8 x} (- 8 x) - e^{- 8 x}) : - e^{- 8 x} x - \frac{1}{8} e^{- 8 x}

= - e^{- 8 x} x - \frac{1}{8} e^{- 8 x}

Add a constant to the solution

= - e^{- 8 x} x - \frac{1}{8} e^{- 8 x} + C

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What is the integral of 8xe^{-8x} ?
The integral of 8xe^{-8x} is -e^{-8x}x-1/8 e^{-8x}+C