Question

The value of \(\int_a^b \frac{|x|}{x} dx\) is

• A. \(|b| - |a|\)
• B. \(|a| - |b|\)
• C. \(|b| + |a|\)
• D. \(-|b| - |a|\)

Hint:

Split the integral at \(x=0\) based on the sign of \(x\).

Answer 1

The Correct Option is A

Step 1: Formula / Definition}
\[ \frac{|x|}{x} = \begin{cases} 1, & x>0 \\ -1, & x<0 \end{cases} \]
Step 2: Calculation / Simplification}
Case 1: \(0 \leq a<b\): \(\int_a^b 1 dx = b - a = |b| - |a|\)
Case 2: \(a<b \leq 0\): \(\int_a^b (-1) dx = -(b - a) = a - b = |b| - |a|\)
Case 3: \(a<0<b\): \(\int_a^0 (-1)dx + \int_0^b 1 dx = -(-a) + b = a + b = |b| - |a|\)
All cases yield \(|b| - |a|\).
Step 3: Final Answer
\[ |b| - |a| \]