Class CBSE Class 12 Mathematics Integrals Q #1299
KNOWLEDGE BASED
UNDERSTAND
3 Marks 2024 AISSCE(Board Exam) SA
Find: $\int\frac{x^{2}}{(x^{2}+4)(x^{2}+9)}dx$
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Detailed Solution

Step 1: Partial Fraction Decomposition

Let $x^2 = y$. Then the integrand becomes $\frac{y}{(y+4)(y+9)}$. We can decompose this into partial fractions:

Step 2: Setting up the equation

$$\frac{y}{(y+4)(y+9)} = \frac{A}{y+4} + \frac{B}{y+9}$$

Step 3: Solving for A and B

Multiplying both sides by $(y+4)(y+9)$, we get: $$y = A(y+9) + B(y+4)$$ To find $A$, let $y = -4$: $$-4 = A(-4+9) + B(-4+4) \Rightarrow -4 = 5A \Rightarrow A = -\frac{4}{5}$$ To find $B$, let $y = -9$: $$-9 = A(-9+9) + B(-9+4) \Rightarrow -9 = -5B \Rightarrow B = \frac{9}{5}$$

Step 4: Substituting A and B

So, we have: $$\frac{y}{(y+4)(y+9)} = -\frac{4}{5(y+4)} + \frac{9}{5(y+9)}$$ Substituting $y = x^2$ back, we get: $$\frac{x^2}{(x^2+4)(x^2+9)} = -\frac{4}{5(x^2+4)} + \frac{9}{5(x^2+9)}$$

Step 5: Integrating

Now, we integrate: $$\int \frac{x^2}{(x^2+4)(x^2+9)} dx = \int \left(-\frac{4}{5(x^2+4)} + \frac{9}{5(x^2+9)}\right) dx$$ $$= -\frac{4}{5} \int \frac{1}{x^2+4} dx + \frac{9}{5} \int \frac{1}{x^2+9} dx$$ We know that $\int \frac{1}{x^2+a^2} dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$. Therefore, $$= -\frac{4}{5} \cdot \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) + \frac{9}{5} \cdot \frac{1}{3} \tan^{-1}\left(\frac{x}{3}\right) + C$$ $$= -\frac{2}{5} \tan^{-1}\left(\frac{x}{2}\right) + \frac{3}{5} \tan^{-1}\left(\frac{x}{3}\right) + C$$

Final Answer: $-\frac{2}{5} \tan^{-1}\left(\frac{x}{2}\right) + \frac{3}{5} \tan^{-1}\left(\frac{x}{3}\right) + C$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concept of partial fraction decomposition and apply the standard integration formula.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific procedure (partial fraction decomposition) and then use a standard integration formula.<\/span>
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of integration techniques, specifically partial fractions, as covered in the textbook.

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