What is the integral of ((e^x+1)^3)/(e^{2x)} ?

The integral of ((e^x+1)^3)/(e^{2x)} is e^x+3x-3/(e^x)-1/2 e^{-2x}+C

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integral of ((e^x+1)^3)/(e^{2x)}

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

e^{x} + 3 x - \frac{3}{e ^{x}} - \frac{1}{2} e^{- 2 x} + C

Solution steps

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

Expand

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

= \int e^{x} + 3 + \frac{3}{e ^{x}} + \frac{1}{e ^{2 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 3 d x + \int \frac{3}{e ^{x}} d x + \int \frac{1}{e ^{2 x}} d x

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

\int 3 d x = 3 x

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

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

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

Add a constant to the solution

= e^{x} + 3 x - \frac{3}{e ^{x}} - \frac{1}{2} e^{- 2 x} + C

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What is the integral of ((e^x+1)^3)/(e^{2x)} ?
The integral of ((e^x+1)^3)/(e^{2x)} is e^x+3x-3/(e^x)-1/2 e^{-2x}+C