What is the integral of (6x+3)e^{-2x} ?

The integral of (6x+3)e^{-2x} is -3e^{-2x}x-3e^{-2x}+C

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integral of (6x+3)e^{-2x}

\int (6 x + 3) e^{- 2 x} d x

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

Solution steps

\int (6 x + 3) e^{- 2 x} d x

Apply Integration By Parts

= - \frac{1}{2} e^{- 2 x} (6 x + 3) - \int - 3 e^{- 2 x} d x

\int - 3 e^{- 2 x} d x = \frac{3}{2} e^{- 2 x}

= - \frac{1}{2} e^{- 2 x} (6 x + 3) - \frac{3}{2} e^{- 2 x}

Simplify

- \frac{1}{2} e^{- 2 x} (6 x + 3) - \frac{3}{2} e^{- 2 x} : - 3 e^{- 2 x} x - 3 e^{- 2 x}

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

Add a constant to the solution

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

What is the integral of (6x+3)e^{-2x} ?
The integral of (6x+3)e^{-2x} is -3e^{-2x}x-3e^{-2x}+C