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

The integral of x/((4+2x)) is 1/2 (2+x-2ln(2|x+2|))+C

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

integral

\frac{x}{( 4 + 2 x )}

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

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Expand

\frac{u - 4}{u} : 1 - \frac{4}{u}

= \frac{1}{4} \cdot \int 1 - \frac{4}{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

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

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= \frac{1}{4} (u - 4 ln ∣ u ∣)

Substitute back

u = 4 + 2 x

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of x/((4+2x)) ?
The integral of x/((4+2x)) is 1/2 (2+x-2ln(2|x+2|))+C