IF `0 lt a lt b` then `int_a^b |x|/x` dx is equal to

Options

A. A. `|b|-|a|`
B. B. `|a|-|b|`
C. C. `|b|`
D. D. 0

Answer

To solve the integral \(\int_a^b \frac{|x|}{x} \, dx\) given that \(0 < a < b\), we can follow these steps:

### Step 1: Analyze the expression \(\frac{|x|}{x}\)
Since both \(a\) and \(b\) are greater than 0, the expression \(|x|\) simplifies to \(x\) for all \(x\) in the interval \([a, b]\). Thus, we can rewrite the integrand:

\[ \frac{|x|}{x} = \frac{x}{x} = 1 \quad \text{for } x > 0 \]

### Step 2: Rewrite the integral
Now we can rewrite the integral as:

\[ \int_a^b \frac{|x|}{x} \, dx = \int_a^b 1 \, dx \]

### Step 3: Evaluate the integral
The integral of 1 with respect to \(x\) over the interval \([a, b]\) is simply the length of the interval:

\[ \int_a^b 1 \, dx = x \bigg|_a^b = b - a \]

### Step 4: Final result
Thus, the value of the integral is:

\[ \int_a^b \frac{|x|}{x} \, dx = b - a \]