What is the integral of (e^x)/(2+e^x) ?

The integral of (e^x)/(2+e^x) is ln|2+e^x|+C

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

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

ln ∣ 2 + e^{x} ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = 2 + e^{x}

= ln ∣ 2 + e^{x} ∣

Add a constant to the solution

= ln ∣ 2 + e^{x} ∣ + C

\frac{\partial}{\partial x} (x e^{2 x y})

x y^{2} \frac{d y}{d x} = y^{3} - x^{3}, y (1) = 1

f (x) - 2 x^{2}

\frac{\partial}{\partial x} (x^{2} y - x y^{2})

f (x) = 2 x + 3, (3, 3)

What is the integral of (e^x)/(2+e^x) ?
The integral of (e^x)/(2+e^x) is ln|2+e^x|+C