What is the integral of (2x+1)/(e^{3x)} ?

The integral of (2x+1)/(e^{3x)} is -2/3 e^{-3x}x-5/9 e^{-3x}+C

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integral of (2x+1)/(e^{3x)}

\int \frac{2 x + 1}{e ^{3 x}} d x

- \frac{2}{3} e^{- 3 x} x - \frac{5}{9} e^{- 3 x} + C

Solution steps

\int \frac{2 x + 1}{e ^{3 x}} d x

Apply Integration By Parts

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

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

\int - \frac{2}{3} e^{- 3 x} d x = \frac{2}{9} e^{- 3 x}

= - \frac{1}{3} e^{- 3 x} (2 x + 1) - \frac{2}{9} e^{- 3 x}

Simplify

- \frac{1}{3} e^{- 3 x} (2 x + 1) - \frac{2}{9} e^{- 3 x} : - \frac{2}{3} e^{- 3 x} x - \frac{5}{9} e^{- 3 x}

= - \frac{2}{3} e^{- 3 x} x - \frac{5}{9} e^{- 3 x}

Add a constant to the solution

= - \frac{2}{3} e^{- 3 x} x - \frac{5}{9} e^{- 3 x} + C

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What is the integral of (2x+1)/(e^{3x)} ?
The integral of (2x+1)/(e^{3x)} is -2/3 e^{-3x}x-5/9 e^{-3x}+C