What is the integral of |2x-6| ?

The integral of |2x-6| is 1/2 |2x-6|(x-3)+C

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integral of |2x-6|

\int ∣ 2 x - 6 ∣ d x

\frac{1}{2} ∣ 2 x - 6 ∣ (x - 3) + C

Solution steps

\int ∣ 2 x - 6 ∣ d x

Apply u-substitution

= \int ∣ u ∣ \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int ∣ u ∣ d u

Use the common integral:

\int ∣ u ∣ d u = \frac{u ∣ u ∣}{2}

= \frac{1}{2} \cdot \frac{u ∣ u ∣}{2}

Substitute back

u = 2 x - 6

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

\frac{( 2 x - 6 ) ∣ 2 x - 6 ∣}{2} = ∣ 2 x - 6 ∣ (x - 3)

= \frac{1}{2} ∣ 2 x - 6 ∣ (x - 3)

Add a constant to the solution

= \frac{1}{2} ∣ 2 x - 6 ∣ (x - 3) + C