What is the integral of 9/(x+2) ?

The integral of 9/(x+2) is 9ln|x+2|+C

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integral of 9/(x+2)

\int \frac{9}{x + 2} d x

9 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{9}{x + 2} d x

Take the constant out:

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

= 9 \cdot \int \frac{1}{x + 2} d x

Apply u-substitution

= 9 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 9 ln ∣ u ∣

Substitute back

u = x + 2

= 9 ln ∣ x + 2 ∣

Add a constant to the solution

= 9 ln ∣ x + 2 ∣ + C