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

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

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

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

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

Solution steps

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

Expand

\frac{1 + 2 e ^{x}}{e ^{2 x}} : \frac{1}{e ^{2 x}} + \frac{2}{e ^{x}}

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

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

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

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

Add a constant to the solution

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

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