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

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

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

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

2 + e^{x} - 2 ln ∣ 2 + e^{x} ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 2}{u} d u

= \int \frac{u - 2}{u} d u

Expand

\frac{u - 2}{u} : 1 - \frac{2}{u}

= \int 1 - \frac{2}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int \frac{2}{u} d u

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= u - 2 ln ∣ u ∣

Substitute back

u = 2 + e^{x}

= 2 + e^{x} - 2 ln ∣ 2 + e^{x} ∣

Add a constant to the solution

= 2 + e^{x} - 2 ln ∣ 2 + e^{x} ∣ + C

\frac{1}{4} x^{2} - \frac{1}{2} ln (x)

\frac{\partial}{\partial x} (x^{2} - 8 x + y^{2} \cdot ln (x - 1))

\int x^{- a} d x

x \to 1 lim (\frac{x ^{3} - x}{x - 1})

x \to 0 lim (- \frac{1}{3 x ^{3}})

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