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

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

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

\int \frac{6 + e ^{3 x}}{e ^{2 x}} d x

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

Solution steps

\int \frac{6 + e ^{3 x}}{e ^{2 x}} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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