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

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

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

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

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

Solution steps

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

Expand

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

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

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

\int 2 d x = 2 x

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

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

Add a constant to the solution

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

y = 4 - x^{2}, y = 2 - x

\frac{d}{d x} (x csc (2 x))

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

y = (5 x - 3)^{4} (2 x - 7)^{- 3}

\int \frac{5}{27 ( x + 3 )} d x

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