What is the integral of (6x)/(e^{3x)} ?

The integral of (6x)/(e^{3x)} is 6(-1/3 e^{-3x}x-1/9 e^{-3x})+C

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integral of (6x)/(e^{3x)}

\int \frac{6 x}{e ^{3 x}} d x

6 (- \frac{1}{3} e^{- 3 x} x - \frac{1}{9} e^{- 3 x}) + C

Solution steps

\int \frac{6 x}{e ^{3 x}} d x

Take the constant out:

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

= 6 \cdot \int \frac{x}{e ^{3 x}} d x

Apply Integration By Parts

: - \frac{1}{3} e^{- 3 x} x - \int - \frac{1}{3} e^{- 3 x} d x

= 6 (- \frac{1}{3} e^{- 3 x} x - \int - \frac{1}{3} e^{- 3 x} d x)

\int - \frac{1}{3} e^{- 3 x} d x = \frac{1}{9} e^{- 3 x}

= 6 (- \frac{1}{3} e^{- 3 x} x - \frac{1}{9} e^{- 3 x})

Add a constant to the solution

= 6 (- \frac{1}{3} e^{- 3 x} x - \frac{1}{9} e^{- 3 x}) + C

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What is the integral of (6x)/(e^{3x)} ?
The integral of (6x)/(e^{3x)} is 6(-1/3 e^{-3x}x-1/9 e^{-3x})+C