What is the integral of (11)/(11+e^x) ?

The integral of (11)/(11+e^x) is x-ln|e^x+11|+C

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integral of (11)/(11+e^x)

\int \frac{11}{11 + e ^{x}} d x

x - ln ∣ e^{x} + 11 ∣ + C

Solution steps

\int \frac{11}{11 + e ^{x}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 11 \cdot \int \frac{1}{11 + e ^{x}} d x

Apply u-substitution

: \int \frac{1}{u ( 11 + u )} d u

= 11 \cdot \int \frac{1}{u ( 11 + u )} d u

Take the partial fraction of

\frac{1}{u ( 11 + u )} : \frac{1}{11 u} - \frac{1}{11 ( u + 11 )}

= 11 \cdot \int \frac{1}{11 u} - \frac{1}{11 ( u + 11 )} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 11 (\int \frac{1}{11 u} d u - \int \frac{1}{11 ( u + 11 )} d u)

\int \frac{1}{11 u} d u = \frac{1}{11} ln ∣ u ∣

\int \frac{1}{11 ( u + 11 )} d u = \frac{1}{11} ln ∣ u + 11 ∣

= 11 (\frac{1}{11} ln ∣ u ∣ - \frac{1}{11} ln ∣ u + 11 ∣)

Substitute back

u = e^{x}

= 11 (\frac{1}{11} ln ∣ e^{x} ∣ - \frac{1}{11} ln ∣ e^{x} + 11 ∣)

Simplify

11 (\frac{1}{11} ln ∣ e^{x} ∣ - \frac{1}{11} ln ∣ e^{x} + 11 ∣) : x - ln ∣ e^{x} + 11 ∣

= x - ln ∣ e^{x} + 11 ∣

Add a constant to the solution

= x - ln ∣ e^{x} + 11 ∣ + C

\frac{d}{d t} (tan (e^{2 t}) + e^{t a n (2 t)})

x \to 0 - lim (\frac{1}{x} - \frac{1}{∣ x ∣})

\frac{d}{d x} (x^{3} - 2 x^{2} + x + 1)

\frac{\partial}{\partial x} (\frac{e ^{x}}{y + x ^{3}})

\frac{d}{d x} (tan (\frac{π}{4}))

What is the integral of (11)/(11+e^x) ?
The integral of (11)/(11+e^x) is x-ln|e^x+11|+C