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

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

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

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

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

Solution steps

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

Expand

\frac{3 x + 1}{e ^{3 x}} : \frac{3 x}{e ^{3 x}} + \frac{1}{e ^{3 x}}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= - e^{- 3 x} x - \frac{1}{3} e^{- 3 x} - \frac{1}{3} e^{- 3 x}

Add similar elements:

- \frac{1}{3} e^{- 3 x} - \frac{1}{3} e^{- 3 x} = - \frac{2}{3} e^{- 3 x}

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

Add a constant to the solution

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

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