What is the integral of 35xe^{-7x} ?

The integral of 35xe^{-7x} is -5e^{-7x}x-5/7 e^{-7x}+C

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integral of 35xe^{-7x}

\int 35 x e^{- 7 x} d x

- 5 e^{- 7 x} x - \frac{5}{7} e^{- 7 x} + C

Solution steps

\int 35 x e^{- 7 x} d x

Take the constant out:

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

= 35 \cdot \int x e^{- 7 x} d x

Apply u-substitution

= 35 \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

= 35 \cdot \frac{1}{49} \cdot \int e^{u} u d u

Apply Integration By Parts

= 35 \cdot \frac{1}{49} (e^{u} u - \int e^{u} d u)

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

= 35 \cdot \frac{1}{49} (e^{u} u - e^{u})

Substitute back

u = - 7 x

= 35 \cdot \frac{1}{49} (e^{- 7 x} (- 7 x) - e^{- 7 x})

Simplify

35 \cdot \frac{1}{49} (e^{- 7 x} (- 7 x) - e^{- 7 x}) : - 5 e^{- 7 x} x - \frac{5}{7} e^{- 7 x}

= - 5 e^{- 7 x} x - \frac{5}{7} e^{- 7 x}

Add a constant to the solution

= - 5 e^{- 7 x} x - \frac{5}{7} e^{- 7 x} + C

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What is the integral of 35xe^{-7x} ?
The integral of 35xe^{-7x} is -5e^{-7x}x-5/7 e^{-7x}+C