What is the integral of ((6+e^x)^2)/(e^x) ?

The integral of ((6+e^x)^2)/(e^x) is -(36)/(e^x)+12x+e^x+C

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integral of ((6+e^x)^2)/(e^x)

\int \frac{( 6 + e ^{x} ) ^{2}}{e ^{x}} d x

- \frac{36}{e ^{x}} + 12 x + e^{x} + C

Solution steps

\int \frac{( 6 + e ^{x} ) ^{2}}{e ^{x}} d x

Expand

\frac{( 6 + e ^{x} ) ^{2}}{e ^{x}} : \frac{36}{e ^{x}} + 12 + e^{x}

= \int \frac{36}{e ^{x}} + 12 + 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{36}{e ^{x}} d x + \int 12 d x + \int e^{x} d x

\int \frac{36}{e ^{x}} d x = - \frac{36}{e ^{x}}

\int 12 d x = 12 x

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

= - \frac{36}{e ^{x}} + 12 x + e^{x}

Add a constant to the solution

= - \frac{36}{e ^{x}} + 12 x + e^{x} + C

f (x) = e^{x} + ln (x + 1)

\frac{1}{2 x ^{\frac{1}{2}}} + \frac{1}{4 x ^{\frac{3}{4}}}

k = 1 \sum \infty k^{- 2.4}

6 x^{2} + \frac{768}{x}

x \to + 1 lim (\frac{- 1}{x - 1})

What is the integral of ((6+e^x)^2)/(e^x) ?
The integral of ((6+e^x)^2)/(e^x) is -(36)/(e^x)+12x+e^x+C