Find the Integral |x|

Problem

(∫_^)(|x|*d(x))

Solution

1. Identify the definition of the absolute value function, which is defined piecewise as x for x≥0 and −x for x<0

2. Integrate the function for the case where x≥0 which yields (∫_^)(x*d(x))=(x2)/2+C

3. Integrate the function for the case where x<0 which yields (∫_^)(−x*d(x))=−(x2)/2+C

4. Combine these cases into a single expression using the relationship |x|=√(,x2) or by noting that x*|x| produces the correct signs for both intervals.

5. Simplify the result to the standard form involving the absolute value notation.

Final Answer

(∫_^)(|x|*d(x))=(x*|x|)/2+C

Want more problems? Check here!