What is the integral of |4x| ?

The integral of |4x| is 2x|x|+C

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integral of |4x|

\int ∣ 4 x ∣ d x

2 x ∣ x ∣ + C

Solution steps

\int ∣ 4 x ∣ d x

Apply absolute rule:

∣ a x ∣ = a ∣ x ∣, a \geq 0

= \int 4 ∣ x ∣ d x

Take the constant out:

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

= 4 \cdot \int ∣ x ∣ d x

Use the common integral:

\int ∣ x ∣ d x = \frac{x ∣ x ∣}{2}

= 4 \cdot \frac{x ∣ x ∣}{2}

Simplify

4 \cdot \frac{x ∣ x ∣}{2} : 2 x ∣ x ∣

= 2 x ∣ x ∣

Add a constant to the solution

= 2 x ∣ x ∣ + C

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