Class CBSE Class 12 Mathematics Integrals Q #902
KNOWLEDGE BASED
APPLY
3 Marks 2023 SA
Evaluate: $\int_{1}^{3}\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}}dx$
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Step-by-Step Solution

**Step 1: Define the integral** Let $I = \int_{1}^{3}\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}}dx$
**Step 2: Apply the property of definite integrals** Using the property $\int_{a}^{b}f(x)dx = \int_{a}^{b}f(a+b-x)dx$, we have: $I = \int_{1}^{3}\frac{\sqrt{4-(4-x)}}{\sqrt{4-x}+\sqrt{4-(4-x)}}dx = \int_{1}^{3}\frac{\sqrt{x}}{\sqrt{4-x}+\sqrt{x}}dx$
**Step 3: Add the two expressions for I** Adding the original and transformed integrals: $2I = \int_{1}^{3}\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}}dx + \int_{1}^{3}\frac{\sqrt{x}}{\sqrt{4-x}+\sqrt{x}}dx = \int_{1}^{3}\frac{\sqrt{4-x}+\sqrt{x}}{\sqrt{x}+\sqrt{4-x}}dx$ $2I = \int_{1}^{3}1 dx$
**Step 4: Evaluate the integral** $2I = [x]_{1}^{3} = 3 - 1 = 2$
**Step 5: Solve for I** $I = \frac{2}{2} = 1$

Correct Answer: 1

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the properties of definite integrals to solve the problem. Specifically, they need to use the property $\int_{a}^{b}f(x)dx = \int_{a}^{b}f(a+b-x)dx$ and then perform algebraic manipulation to arrive at the solution.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure involving the application of a definite integral property and algebraic simplification to evaluate the integral.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge and application of definite integral properties, a standard topic covered in the textbook.

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