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

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

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

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

1 + e^{x} - ln ∣ 1 + e^{x} ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 1}{u} d u

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

Expand

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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= u - ln ∣ u ∣

Substitute back

u = 1 + e^{x}

= 1 + e^{x} - ln ∣ 1 + e^{x} ∣

Add a constant to the solution

= 1 + e^{x} - ln ∣ 1 + e^{x} ∣ + C

\frac{d}{d x} \frac{( x ^{2} + 32 x ^{\frac{1}{2}} )}{8}

(e^{y} + 1)^{2} e^{- y} d x + (e^{x} + 1) e^{- x} d y = 0

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

x \to 1 lim (\frac{2 - 1}{x ^{3} - 3})

\frac{x ^{3}}{3 - x ^{2}}

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