What is the integral of (e^{5x})/(1+e^{5x)} ?

The integral of (e^{5x})/(1+e^{5x)} is 1/5 ln|1+e^{5x}|+C

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integral of (e^{5x})/(1+e^{5x)}

\int \frac{e ^{5 x}}{1 + e ^{5 x}} d x

\frac{1}{5} ln 1 + e^{5 x} + C

Solution steps

\int \frac{e ^{5 x}}{1 + e ^{5 x}} d x

Apply u-substitution

: \int \frac{1}{5 u} d u

= \int \frac{1}{5 u} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = 1 + e^{5 x}

= \frac{1}{5} ln 1 + e^{5 x}

Add a constant to the solution

= \frac{1}{5} ln 1 + e^{5 x} + C

\int (x^{2} y^{2}) d x

x^{2} - 5 x + \frac{1}{x ^{\frac{3}{2}}}

x \to 3 lim (\frac{x - 3}{1 + x})

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

\frac{2}{3} x^{3} + 4 x^{2} + 6 x - 5

What is the integral of (e^{5x})/(1+e^{5x)} ?
The integral of (e^{5x})/(1+e^{5x)} is 1/5 ln|1+e^{5x}|+C