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Popular >

integral of 1/(1+e^{-t)}

Solution

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

Solution

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

Simplify

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

Add a constant to the solution

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

Popular Examples

integral of (x^2)/(x^2+81)

\int \frac{x ^{2}}{x ^{2} + 81} d x

(y^'-e^{-t}+5)/y =-5,y(0)=-4

\frac{y ^{^{'}} - e ^{- t} + 5}{y} = - 5, y (0) = - 4

slope of (5,-1),(-3,4)slope

(5, - 1), (- 3, 4)

derivative of (x-3/(x^2-3x+2))

\frac{d}{d x} (\frac{x - 3}{x ^{2} - 3 x + 2})

integral of 4y^4

\int 4 y^{4} d y

Frequently Asked Questions (FAQ)

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