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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3 ln ( 3 )} ln ∣ u ∣

Substitute back

u = 3^{3 x + 2} - 3

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

Add a constant to the solution

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

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