Find the Antiderivative |x|

Problem

(∫_^)(|x|*d(x))

Solution

1. Define the absolute value function as a piecewise function to handle the different signs of x

|x|={[x,if *x≥0],[−x,if *x<0])

2. Integrate each piece of the function separately using the power rule for integration.

(∫_^)(x*d(x))=(x2)/2+(C_1)

(∫_^)(−x*d(x))=−(x2)/2+(C_2)

3. Combine the results into a single expression. Since (x2)/2 can be written as (x⋅x)/2 and |x| matches the sign of x the antiderivative can be expressed using the absolute value notation.

(x*|x|)/2+C

4. Verify the result by differentiating. The derivative of (x*|x|)/2 is |x| for all x≠0 and the function is continuous at x=0

d()/d(x)(x*|x|)/2=|x|

Final Answer

(∫_^)(|x|*d(x))=(x*|x|)/2+C

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