What is the integral of 14xe^{-7x} ?

The integral of 14xe^{-7x} is -2e^{-7x}x-2/7 e^{-7x}+C

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integral of 14xe^{-7x}

\int 14 x e^{- 7 x} d x

- 2 e^{- 7 x} x - \frac{2}{7} e^{- 7 x} + C

Solution steps

\int 14 x e^{- 7 x} d x

Take the constant out:

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

= 14 \cdot \int x e^{- 7 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{49} d u

= 14 \cdot \int \frac{e ^{u} u}{49} d u

Take the constant out:

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

= 14 \cdot \frac{1}{49} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 14 \cdot \frac{1}{49} (e^{u} u - \int e^{u} d u)

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

= 14 \cdot \frac{1}{49} (e^{u} u - e^{u})

Substitute back

u = - 7 x

= 14 \cdot \frac{1}{49} (e^{- 7 x} (- 7 x) - e^{- 7 x})

Simplify

14 \cdot \frac{1}{49} (e^{- 7 x} (- 7 x) - e^{- 7 x}) : - 2 e^{- 7 x} x - \frac{2}{7} e^{- 7 x}

= - 2 e^{- 7 x} x - \frac{2}{7} e^{- 7 x}

Add a constant to the solution

= - 2 e^{- 7 x} x - \frac{2}{7} e^{- 7 x} + C

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What is the integral of 14xe^{-7x} ?
The integral of 14xe^{-7x} is -2e^{-7x}x-2/7 e^{-7x}+C