What is the integral of |2x+2| ?

The integral of |2x+2| is |x+1|(x+1)+C

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integral of |2x+2|

\int ∣ 2 x + 2 ∣ d x

∣ x + 1 ∣ (x + 1) + C

Solution steps

\int ∣ 2 x + 2 ∣ d x

Factor

∣ 2 x + 2 ∣ : 2 ∣ x + 1 ∣

= \int 2 ∣ x + 1 ∣ d x

Take the constant out:

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

= 2 \cdot \int ∣ x + 1 ∣ d x

Apply u-substitution

: \int ∣ u ∣ d u

= 2 \cdot \int ∣ u ∣ d u

Use the common integral:

\int ∣ u ∣ d u = \frac{u ∣ u ∣}{2}

= 2 \cdot \frac{u ∣ u ∣}{2}

Substitute back

u = x + 1

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

Simplify

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

= ∣ x + 1 ∣ (x + 1)

Add a constant to the solution

= ∣ x + 1 ∣ (x + 1) + C

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