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

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

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

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

- \frac{2}{3} e^{- 3 x} + \frac{1}{2} e^{- 2 x} + C

Solution steps

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

Expand

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

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

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

\int \frac{1}{e ^{2 x}} d x = - \frac{1}{2} e^{- 2 x}

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

Simplify

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

Add a constant to the solution

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

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