Class CBSE Class 12 Mathematics Integrals Q #1248
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2024 AISSCE(Board Exam) VSA
Given $\frac{d}{dx}F(x)=\frac{1}{\sqrt{2x-x^{2}}}$ and $F(1)=0$, find $F(x)$.
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Detailed Solution

Step 1: Recognize the derivative

We are given that $\frac{d}{dx}F(x)=\frac{1}{\sqrt{2x-x^{2}}}$. This means that $F(x)$ is the antiderivative of $\frac{1}{\sqrt{2x-x^{2}}}$.

Step 2: Complete the square

We need to integrate $\frac{1}{\sqrt{2x-x^{2}}}$. First, complete the square in the denominator: $2x - x^2 = -(x^2 - 2x) = -(x^2 - 2x + 1 - 1) = -( (x-1)^2 - 1) = 1 - (x-1)^2$. So, we have $\frac{1}{\sqrt{1 - (x-1)^2}}$.

Step 3: Integrate

Now we integrate: $\int \frac{1}{\sqrt{1 - (x-1)^2}} dx$. Let $u = x-1$, so $du = dx$. Then the integral becomes: $\int \frac{1}{\sqrt{1 - u^2}} du = \arcsin(u) + C = \arcsin(x-1) + C$. Thus, $F(x) = \arcsin(x-1) + C$.

Step 4: Use the initial condition

We are given that $F(1) = 0$. Plugging this into our expression for $F(x)$: $F(1) = \arcsin(1-1) + C = \arcsin(0) + C = 0 + C = 0$. Therefore, $C = 0$.

Step 5: Final Answer

So, $F(x) = \arcsin(x-1)$.

Final Answer: $F(x) = \arcsin(x-1)$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the relationship between a function and its derivative, and how to use the given initial condition to find the specific function.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply the procedure of integration and using initial conditions to find the particular solution. Completing the square is also a procedural skill.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of integration and its application in finding a function given its derivative and an initial condition, a standard topic in the textbook.

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