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

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

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integral of 1/(1+e^t)

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

- ln 1 + e^{t} + t + C

Solution steps

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

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

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

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 ^{t}}{1 + e ^{t}} d t + \int 1 d t

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

\int 1 d t = t

= - ln 1 + e^{t} + t

Add a constant to the solution

= - ln 1 + e^{t} + t + C

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