What is the integral of |2x-4| ?

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

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integral of |2x-4|

\int ∣ 2 x - 4 ∣ d x

\frac{1}{2} x ∣ 2 x - 4 ∣ - ∣ 2 x - 4 ∣ + C

Solution steps

\int ∣ 2 x - 4 ∣ 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 - 4

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

Simplify

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

= \frac{1}{2} x ∣ 2 x - 4 ∣ - ∣ 2 x - 4 ∣

Add a constant to the solution

= \frac{1}{2} x ∣ 2 x - 4 ∣ - ∣ 2 x - 4 ∣ + C

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