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

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

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integral of (1/(1+e^x))

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

- ln ∣ 1 + e^{x} ∣ + x + C

Solution steps

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

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

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

\int \frac{e ^{x}}{1 + e ^{x}} d x = ln ∣ 1 + e^{x} ∣

\int 1 d x = x

= - ln ∣ 1 + e^{x} ∣ + x

Add a constant to the solution

= - ln ∣ 1 + e^{x} ∣ + x + C

x \to \infty lim (- e^{- a x})

\int \frac{x ^{2}}{5 + 2 x ^{3}} d x

\frac{\partial}{\partial x} (sin (2 x + 3 y))

y = 7 x - x^{2}

\int_{- 1}^{1} \frac{10}{x ^{2}} d x

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