What is the integral of 12xe^{-2x} ?

The integral of 12xe^{-2x} is 3(-2e^{-2x}x-e^{-2x})+C

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integral of 12xe^{-2x}

\int 12 x e^{- 2 x} d x

3 (- 2 e^{- 2 x} x - e^{- 2 x}) + C

Solution steps

\int 12 x e^{- 2 x} d x

Take the constant out:

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

= 12 \cdot \int x e^{- 2 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{4} d u

= 12 \cdot \int \frac{e ^{u} u}{4} d u

Take the constant out:

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

= 12 \cdot \frac{1}{4} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 12 \cdot \frac{1}{4} (e^{u} u - \int e^{u} d u)

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

= 12 \cdot \frac{1}{4} (e^{u} u - e^{u})

Substitute back

u = - 2 x

= 12 \cdot \frac{1}{4} (e^{- 2 x} (- 2 x) - e^{- 2 x})

Simplify

12 \cdot \frac{1}{4} (e^{- 2 x} (- 2 x) - e^{- 2 x}) : 3 (- 2 e^{- 2 x} x - e^{- 2 x})

= 3 (- 2 e^{- 2 x} x - e^{- 2 x})

Add a constant to the solution

= 3 (- 2 e^{- 2 x} x - e^{- 2 x}) + C

\frac{d}{d x} ((sin (3 x))^{3} + 5 x)

cos (2 t + π)

ln (sec (x) + tan (x))

\frac{d}{d x} (x^{\frac{1}{n}})

x + y e^{\frac{y}{x}} d x - x e^{\frac{y}{x}} d y = 0

What is the integral of 12xe^{-2x} ?
The integral of 12xe^{-2x} is 3(-2e^{-2x}x-e^{-2x})+C