What is the integral of (3-|x-3|) ?

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

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integral of (3-|x-3|)

\int (3 - ∣ x - 3 ∣) d x

3 x - \frac{( x - 3 ) ∣ x - 3 ∣}{2} + C

Solution steps

\int 3 - ∣ x - 3 ∣ d x

Apply the Sum Rule:

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

= \int 3 d x - \int ∣ x - 3 ∣ d x

\int 3 d x = 3 x

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

= 3 x - \frac{( x - 3 ) ∣ x - 3 ∣}{2}

Add a constant to the solution

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

What is the integral of (3-|x-3|) ?
The integral of (3-|x-3|) is 3x-((x-3)|x-3|)/2+C