Integral of the Absolute Value of \( x \): \( \int |x| \, dx \)

Derivation and Evaluation

Evaluate the indefinite integral:

\[ \int |x| \, dx \]

Rewrite the integrand as a product:

\[ \int |x| \, dx = \int 1 \cdot |x| \, dx \quad (I) \]

Recall that \( |x| = \sqrt{x^2} \) and its derivative with respect to \( x \) is given by:

\[ \dfrac{d(|x|)}{dx} = \dfrac{d(\sqrt{x^2})}{dx} = \dfrac{x}{\sqrt{x^2}} = \dfrac{x}{|x|} \quad (II) \]

Apply the integration by parts formula: \( \displaystyle \int u' v \, dx = u v - \int u v' \, dx \) to the integral on the right side of (I).

Let \( u' = 1 \) and \( v = |x| \), which gives \( u = x \) and \( v' = \dfrac{x}{|x|} \) (from step II).

Substitute these into the integration by parts formula:

\[ = x |x| - \int x \cdot \dfrac{x}{|x|} \, dx \quad (III) \]

Simplify the integrand term \( x \cdot \dfrac{x}{|x|} \):

\[ x \cdot \dfrac{x}{|x|} = \dfrac{x^2}{|x|} = \dfrac{|x|^2}{|x|} = |x| \]

Substitute this simplified integrand back into equation (III):

\[ \int |x| \, dx = x |x| - \int |x| \, dx \]

Add \( \displaystyle \int |x| \, dx \) to both sides:

\[ \int |x| \, dx + \int |x| \, dx = x |x| \] \[ 2 \int |x| \, dx = x |x| \]

Dividing by 2 and adding the constant of integration \( C \), the final answer is given by: