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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Add a constant to the solution

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

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