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

The integral of 6/(4x-2) is 3/2 ln|4x-2|+C

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integral of 6/(4x-2)

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

\frac{3}{2} ln ∣ 4 x - 2 ∣ + C

Solution steps

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

Take the constant out:

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

= 6 \cdot \int \frac{1}{4 x - 2} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 6 \cdot \frac{1}{4} ln ∣ u ∣

Substitute back

u = 4 x - 2

= 6 \cdot \frac{1}{4} ln ∣ 4 x - 2 ∣

Simplify

6 \cdot \frac{1}{4} ln ∣ 4 x - 2 ∣ : \frac{3}{2} ln ∣ 4 x - 2 ∣

= \frac{3}{2} ln ∣ 4 x - 2 ∣

Add a constant to the solution

= \frac{3}{2} ln ∣ 4 x - 2 ∣ + C

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