Class CBSE Class 12 Mathematics Definite Integrals Q #1430
COMPETENCY BASED
APPLY
3 Marks 2025 AISSCE(Board Exam) SA
Evaluate: $\int_{1}^{4}(|x-2|+|x-4|)dx$.
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Detailed Solution

Step 1: Break down the absolute value functions

We need to consider the intervals where the expressions inside the absolute values change signs. $|x-2| = \begin{cases} x-2, & x \geq 2 \\ 2-x, & x < 2 \end{cases}$ $|x-4| = \begin{cases} x-4, & x \geq 4 \\ 4-x, & x < 4 \end{cases}$

Step 2: Split the integral based on the intervals

Since the integral is from 1 to 4, we split the integral at $x=2$. $\int_{1}^{4}(|x-2|+|x-4|)dx = \int_{1}^{2}(|x-2|+|x-4|)dx + \int_{2}^{4}(|x-2|+|x-4|)dx$

Step 3: Evaluate the first integral

For $1 \leq x < 2$, $|x-2| = 2-x$ and $|x-4| = 4-x$. $\int_{1}^{2}(2-x+4-x)dx = \int_{1}^{2}(6-2x)dx = [6x-x^2]_{1}^{2} = (12-4) - (6-1) = 8-5 = 3$

Step 4: Evaluate the second integral

For $2 \leq x \leq 4$, $|x-2| = x-2$ and $|x-4| = 4-x$. $\int_{2}^{4}(x-2+4-x)dx = \int_{2}^{4}2dx = [2x]_{2}^{4} = 8-4 = 4$

Step 5: Combine the results

$\int_{1}^{4}(|x-2|+|x-4|)dx = 3 + 4 = 7$

Final Answer: 7

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of absolute value functions and definite integrals to solve the problem. They must break down the integral into intervals based on where the absolute value expressions change signs and then evaluate each integral separately.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to evaluate the definite integral involving absolute value functions. This includes identifying the intervals, splitting the integral, and applying the correct integration techniques.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concepts of definite integrals and absolute value functions in a problem-solving context, which aligns with competency-based assessment.

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