Question

Int(1 \(1 + {e}^{x} )​

Answer 1

`\begin{aligned}\int{(1)/(1 + e^(x))\cdot dx}= x - \ln(1 + e^(x)) + C\end{aligned}`.

Explanation:

The first derivative of the denominator
`1 + e^(x)` is
`e^(x)`. Rewrite the fraction to obtain that expression on the numerator.

`\begin{aligned}(1)/(1 + e^(x)) &= (1 + e^(x))/(1 + e^(x)) - (e^(x))/(1+e^(x))\\&=1-(e^(x))/(1+e^(x))\end{aligned}`.

In other words,

`\begin{aligned}\int{(1)/(1 + e^(x))\cdot dx} &= \int{dx} - \int{(e^(x))/(1+e^(x))\cdot dx}\end{aligned}`.

Apply
`u`-substitution on the integral
`\displaystyle \int{(e^(x))/(1+e^(x))\cdot dx}`:

Let
`u = 1 + e^(x)`.
`u > 1`.

`du = e^(x)\cdot dx`.

`\displaystyle \int{(e^(x))/(1+e^(x))\cdot dx} = \int{(du)/(u)} = ln(|u|) = ln(u) +C = \ln{(1+e^(x))}+C`.

Therefore

`\begin{aligned}\int{(1)/(1 + e^(x))\cdot dx} &= \int{dx} - \int{(e^(x))/(1+e^(x))\cdot dx}\\ & = x - \ln{(1 + e^(x))}+C\end{aligned}`.