What is the integral of (e^{6x})/(e^{6x)+6} ?

The integral of (e^{6x})/(e^{6x)+6} is 1/6 ln|e^{6x}+6|+C

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integral of (e^{6x})/(e^{6x)+6}

\int \frac{e ^{6 x}}{e ^{6 x} + 6} d x

\frac{1}{6} ln e^{6 x} + 6 + C

Solution steps

\int \frac{e ^{6 x}}{e ^{6 x} + 6} d x

Apply u-substitution

= \int \frac{1}{6 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} ln ∣ u ∣

Substitute back

u = e^{6 x} + 6

= \frac{1}{6} ln e^{6 x} + 6

Add a constant to the solution

= \frac{1}{6} ln e^{6 x} + 6 + C

What is the integral of (e^{6x})/(e^{6x)+6} ?
The integral of (e^{6x})/(e^{6x)+6} is 1/6 ln|e^{6x}+6|+C