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

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

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integral of (3^x)/(3^x+2)

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

\frac{1}{ln ( 3 )} ln ∣ 3^{x} + 2 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{ln ( 3 ) u} d u

= \int \frac{1}{ln ( 3 ) u} d u

Take the constant out:

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

= \frac{1}{ln ( 3 )} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{ln ( 3 )} ln ∣ u ∣

Substitute back

u = 3^{x} + 2

= \frac{1}{ln ( 3 )} ln ∣ 3^{x} + 2 ∣

Add a constant to the solution

= \frac{1}{ln ( 3 )} ln ∣ 3^{x} + 2 ∣ + C

x \to 0 lim (\frac{1}{e ^{\frac{1}{x}}})

\frac{\partial}{\partial z} (\frac{x}{z})

g (x) = ln (x e^{- 7 x})

\frac{\partial}{\partial y} (x^{2} y^{\frac{1}{3}})

\int \frac{u}{u ^{2} + 1} d u

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