What is the integral of (e^{3x})/(e^{3x)+2} ?

The integral of (e^{3x})/(e^{3x)+2} is 1/3 ln|e^{3x}+2|+C

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integral of (e^{3x})/(e^{3x)+2}

\int \frac{e ^{3 x}}{e ^{3 x} + 2} d x

\frac{1}{3} ln e^{3 x} + 2 + C

Solution steps

\int \frac{e ^{3 x}}{e ^{3 x} + 2} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = e^{3 x} + 2

= \frac{1}{3} ln e^{3 x} + 2

Add a constant to the solution

= \frac{1}{3} ln e^{3 x} + 2 + C

\frac{d}{d x} (\frac{x ^{4} + 3}{6 x})

\frac{d}{d y} (3 y)

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

f (y) = 2 x, 0, 4

e^{t} cos (t) + e^{t} sin (t)

What is the integral of (e^{3x})/(e^{3x)+2} ?
The integral of (e^{3x})/(e^{3x)+2} is 1/3 ln|e^{3x}+2|+C