What is the integral of 3/(2x+6) ?

The integral of 3/(2x+6) is 3/2 ln|2x+6|+C

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integral of 3/(2x+6)

\int \frac{3}{2 x + 6} d x

\frac{3}{2} ln ∣ 2 x + 6 ∣ + C

Solution steps

\int \frac{3}{2 x + 6} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{2 x + 6} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= 3 \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = 2 x + 6

= 3 \cdot \frac{1}{2} ln ∣ 2 x + 6 ∣

Simplify

3 \cdot \frac{1}{2} ln ∣ 2 x + 6 ∣ : \frac{3}{2} ln ∣ 2 x + 6 ∣

= \frac{3}{2} ln ∣ 2 x + 6 ∣

Add a constant to the solution

= \frac{3}{2} ln ∣ 2 x + 6 ∣ + C

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