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

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

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

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

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

Solution steps

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

Expand

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

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

\int e^{x} d x = e^{x}

\int 2 d x = 2 x

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

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

Simplify

= e^{x} + 2 x + \frac{2}{e ^{x}}

Add a constant to the solution

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

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