What is the integral of 5.4xe^{-0.1x} ?

The integral of 5.4xe^{-0.1x} is 540(-0.1e^{-0.1x}x-e^{-0.1x})+C

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integral of 5.4xe^{-0.1x}

\int 5.4 x e^{- 0.1 x} d x

540 (- 0.1 e^{- 0.1 x} x - e^{- 0.1 x}) + C

Solution steps

\int 5.4 x e^{- 0.1 x} d x

Take the constant out:

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

= 5.4 \cdot \int x e^{- 0.1 x} d x

Apply u-substitution

: \int \frac{10}{0.1} e^{u} u d u

= 5.4 \cdot \int \frac{10}{0.1} e^{u} u d u

Take the constant out:

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

= 5.4 \cdot \frac{10}{0.1} \cdot \int e^{u} u d u

Simplify

= 5.4 \cdot 100 \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 5.4 \cdot 100 (e^{u} u - \int e^{u} d u)

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

= 5.4 \cdot 100 (e^{u} u - e^{u})

Substitute back

u = - 0.1 x

= 5.4 \cdot 100 (e^{- 0.1 x} (- 0.1 x) - e^{- 0.1 x})

Simplify

= 540 (- 0.1 e^{- 0.1 x} x - e^{- 0.1 x})

Add a constant to the solution

= 540 (- 0.1 e^{- 0.1 x} x - e^{- 0.1 x}) + C

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What is the integral of 5.4xe^{-0.1x} ?
The integral of 5.4xe^{-0.1x} is 540(-0.1e^{-0.1x}x-e^{-0.1x})+C