Question

$\int_{1}^{5} \left(|x - 3| + |1 - x|\right) \, dx =$

• A. $12$
• B. $\frac{5}{6}$
• C. $21$
• D. $10$

Answer 1

The Correct Option is A

1. Understand the problem:

We need to evaluate the definite integral \( \int_{1}^{5} \left( |x - 3| + |1 - x| \right) dx \).

2. Break down the absolute value functions:

The integral contains two absolute value functions: \( |x - 3| \) and \( |1 - x| \). We'll analyze their behavior in the interval [1, 5].

3. Analyze \( |1 - x| \):

Since \( x \in [1, 5] \):

• For \( x \geq 1 \), \( |1 - x| = x - 1 \)

This simplifies to \( (x - 1) \) across the entire interval.

4. Analyze \( |x - 3| \):

The critical point is at \( x = 3 \):

• For \( 1 \leq x \leq 3 \), \( |x - 3| = 3 - x \)
• For \( 3 \leq x \leq 5 \), \( |x - 3| = x - 3 \)

5. Split the integral:

Based on the critical point at \( x = 3 \), we split the integral into two parts:

\[ \int_{1}^{5} \left( |x - 3| + |1 - x| \right) dx = \int_{1}^{3} \left( (3 - x) + (x - 1) \right) dx + \int_{3}^{5} \left( (x - 3) + (x - 1) \right) dx \]

6. Simplify the integrands:

For \( 1 \leq x \leq 3 \):

\[ (3 - x) + (x - 1) = 2 \]

For \( 3 \leq x \leq 5 \):

\[ (x - 3) + (x - 1) = 2x - 4 \]

7. Evaluate the integrals:

First integral (from 1 to 3):

\[ \int_{1}^{3} 2 \, dx = 2(3 - 1) = 4 \]

Second integral (from 3 to 5):

\[ \int_{3}^{5} (2x - 4) dx = \left[ x^2 - 4x \right]_{3}^{5} = (25 - 20) - (9 - 12) = 5 - (-3) = 8 \]

8. Sum the results:

\[ 4 + 8 = 12 \]

Correct Answer: (A) 12

Answer 2

Split the integral based on the behavior of the absolute values: 1. For $x \in [1, 3]$, $|x - 3| = 3 - x$ and $|1 - x| = x - 1$. 2. For $x \in [3, 5]$, $|x - 3| = x - 3$ and $|1 - x| = x - 1$. 

The integral becomes: \[ \int_{1}^{5} \left(|x - 3| + |1 - x|\right) dx = \int_{1}^{3} \left((3 - x) + (x - 1)\right) dx + \int_{3}^{5} \left((x - 3) + (x - 1)\right) dx. \] Simplify each part: 1. For $x \in [1, 3]$: \[ \int_{1}^{3} \left((3 - x) + (x - 1)\right) dx = \int_{1}^{3} (2) dx = 2(3 - 1) = 4. \] 2. For $x \in [3, 5]$: \[ \int_{3}^{5} \left((x - 3) + (x - 1)\right) dx = \int_{3}^{5} (2x - 4) dx = \left[x^2 - 4x\right]_{3}^{5}. \] Evaluate: \[ \left[x^2 - 4x\right]_{3}^{5} = (25 - 20) - (9 - 12) = 5 + 3 = 8. \] Add the results: \[ 4 + 8 = 12. \]