Question

Evaluate the integral: \[ \int_{-1}^{1} \frac{|x|}{x} dx, \, x \neq 0 \]

• A. \( -1 \)
• B. \( 0 \)
• C. \( 1 \)
• D. \( 2 \)

Hint:

When dealing with absolute values, split the integral into intervals where the function behaves simply, either as \( x \) or \( -x \).

Answer 1

The Correct Option is B

The integral to be evaluated is:

\[ \int_{-1}^{1} \frac{|x|}{x}\,dx \]
The absolute value function \(|x|\) requires splitting the integral into two parts: one for \(x > 0\) and another for \(x < 0\). These parts are evaluated separately.

Step 1: Splitting the Integral

For \(x > 0\), \(|x| = x\). Hence,
\[ \int_{0}^{1} \frac{x}{x}\,dx = \int_{0}^{1} 1\,dx \] \[ = 1 \]
For \(x < 0\), \(|x| = -x\). Hence,
\[ \int_{-1}^{0} \frac{-x}{x}\,dx = \int_{-1}^{0} (-1)\,dx \] \[ = -1 \]

Step 2: Combining the Results

\[ \int_{-1}^{1} \frac{|x|}{x}\,dx = \int_{-1}^{0} (-1)\,dx + \int_{0}^{1} 1\,dx = -1 + 1 = 0 \]

Final Answer:
The value of the integral is 0.