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

The integral of ((2+e^x)^2)/(e^x) is -4/(e^x)+4x+e^x+C

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

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

- \frac{4}{e ^{x}} + 4 x + e^{x} + C

Solution steps

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

Expand

\frac{( 2 + e ^{x} ) ^{2}}{e ^{x}} : \frac{4}{e ^{x}} + 4 + e^{x}

= \int \frac{4}{e ^{x}} + 4 + 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{4}{e ^{x}} d x + \int 4 d x + \int e^{x} d x

\int \frac{4}{e ^{x}} d x = - \frac{4}{e ^{x}}

\int 4 d x = 4 x

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

= - \frac{4}{e ^{x}} + 4 x + e^{x}

Add a constant to the solution

= - \frac{4}{e ^{x}} + 4 x + e^{x} + C

What is the integral of ((2+e^x)^2)/(e^x) ?
The integral of ((2+e^x)^2)/(e^x) is -4/(e^x)+4x+e^x+C