What is the integral of (4x)/(2x+6) ?

The integral of (4x)/(2x+6) is 2x+6-6ln|2x+6|+C

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integral of (4x)/(2x+6)

\int \frac{4 x}{2 x + 6} d x

2 x + 6 - 6 ln ∣ 2 x + 6 ∣ + C

Solution steps

\int \frac{4 x}{2 x + 6} d x

Take the constant out:

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

= 4 \cdot \int \frac{x}{2 x + 6} d x

Apply u-substitution

= 4 \cdot \int \frac{u - 6}{4 u} d u

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \int \frac{u - 6}{u} d u

Expand

\frac{u - 6}{u} : 1 - \frac{6}{u}

= 4 \cdot \frac{1}{4} \cdot \int 1 - \frac{6}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 \cdot \frac{1}{4} (\int 1 d u - \int \frac{6}{u} d u)

\int 1 d u = u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

= 4 \cdot \frac{1}{4} (u - 6 ln ∣ u ∣)

Substitute back

u = 2 x + 6

= 4 \cdot \frac{1}{4} (2 x + 6 - 6 ln ∣ 2 x + 6 ∣)

Simplify

4 \cdot \frac{1}{4} (2 x + 6 - 6 ln ∣ 2 x + 6 ∣) : 2 x + 6 - 6 ln ∣ 2 x + 6 ∣

= 2 x + 6 - 6 ln ∣ 2 x + 6 ∣

Add a constant to the solution

= 2 x + 6 - 6 ln ∣ 2 x + 6 ∣ + C

What is the integral of (4x)/(2x+6) ?
The integral of (4x)/(2x+6) is 2x+6-6ln|2x+6|+C