What is the integral of (e^x)/(6e^x+2) ?

The integral of (e^x)/(6e^x+2) is 1/6 ln|6e^x+2|+C

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integral of (e^x)/(6e^x+2)

\int \frac{e ^{x}}{6 e ^{x} + 2} d x

\frac{1}{6} ln ∣ 6 e^{x} + 2 ∣ + C

Solution steps

\int \frac{e ^{x}}{6 e ^{x} + 2} 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 = 6 e^{x} + 2

= \frac{1}{6} ln ∣ 6 e^{x} + 2 ∣

Add a constant to the solution

= \frac{1}{6} ln ∣ 6 e^{x} + 2 ∣ + C

What is the integral of (e^x)/(6e^x+2) ?
The integral of (e^x)/(6e^x+2) is 1/6 ln|6e^x+2|+C