Question

\( \int_{-2}^{2} \frac{|x-3|}{x-3} dx = \)

• A. \( -4 \)
• B. \( -2 \)
• C. \( 0 \)
• D. \( 2 \)

Hint:

Always simplify modulus expressions firstCheck sign of expression in given interval before integrating.

Answer 1

The Correct Option is A

Step 1: Analyze the integrand.
\[ \frac{|x-3|}{x-3} = \begin{cases} 1, & x>3 -1, & x<3 \end{cases} \]

Step 2: Check interval.
Integration limits are:
\[ -2 \text{ to } 2 \]
Here, all values satisfy \( x<3 \).

Step 3: Simplify integrand.
So:
\[ \frac{|x-3|}{x-3} = -1 \]

Step 4: Substitute in integral.
\[ \int_{-2}^{2} -1 \, dx \]

Step 5: Integrate.
\[ = - \int_{-2}^{2} dx \]
\[ = -(2 - (-2)) \]
\[ = -4 \]

Step 6: Interpretation.
Constant function integration gives interval length multiplied by constant.

Step 7: Final conclusion.
\[ \boxed{-4} \]