What is the integral of (e^x)/(1+3e^{-x)} ?

The integral of (e^x)/(1+3e^{-x)} is e^x+3-3ln|e^x+3|+C

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integral of (e^x)/(1+3e^{-x)}

\int \frac{e ^{x}}{1 + 3 e ^{- x}} d x

e^{x} + 3 - 3 ln ∣ e^{x} + 3 ∣ + C

Solution steps

\int \frac{e ^{x}}{1 + 3 e ^{- x}} d x

Apply u-substitution

: \int \frac{u}{u + 3} d u

= \int \frac{u}{u + 3} d u

Apply u-substitution

: \int \frac{v - 3}{v} d v

= \int \frac{v - 3}{v} d v

Expand

\frac{v - 3}{v} : 1 - \frac{3}{v}

= \int 1 - \frac{3}{v} d v

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 v - \int \frac{3}{v} d v

\int 1 d v = v

\int \frac{3}{v} d v = 3 ln ∣ v ∣

= v - 3 ln ∣ v ∣

Substitute back

= e^{x} + 3 - 3 ln ∣ e^{x} + 3 ∣

Add a constant to the solution

= e^{x} + 3 - 3 ln ∣ e^{x} + 3 ∣ + C

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What is the integral of (e^x)/(1+3e^{-x)} ?
The integral of (e^x)/(1+3e^{-x)} is e^x+3-3ln|e^x+3|+C