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

We are asked to evaluate the integral: \[ \int_{-1}^{1} \frac{|x|}{x} dx \] The absolute value function \( |x| \) splits the integral into two parts: one for \( x>0 \) and another for \( x<0 \). We will evaluate these parts separately. Step 1: Split the integral at 0 For \( x>0 \), \( |x| = x \), so the integral becomes: \[ \int_{0}^{1} \frac{x}{x} dx = \int_{0}^{1} 1 dx \] This integral is straightforward: \[ \int_{0}^{1} 1 dx = 1 \] For \( x<0 \), \( |x| = -x \), so the integral becomes: \[ \int_{-1}^{0} \frac{-x}{x} dx = \int_{-1}^{0} (-1) dx \] This integral evaluates to: \[ \int_{-1}^{0} (-1) dx = -1 \] Step 2: Combine the results Now, adding the results from both parts of the integral: \[ \int_{-1}^{1} \frac{|x|}{x} dx = \int_{-1}^{0} (-1) dx + \int_{0}^{1} 1 dx = -1 + 1 = 0 \] Thus, the value of the integral is \( 0 \).