Q1255: Find: $\int\frac{x^{2}+1}{(x^{2}+2)(x^{2}+4)}dx$...

Class CBSE Class 12 Mathematics Integrals Q #1255

KNOWLEDGE BASED

UNDERSTAND

3 Marks 2024 AISSCE(Board Exam) SA

Find: $\int\frac{x^{2}+1}{(x^{2}+2)(x^{2}+4)}dx$

Prev Next

AI Tutor Explanation

Detailed Solution

Step 1: Partial Fraction Decomposition

Let $x^2 = y$. Then the integrand becomes $\frac{y+1}{(y+2)(y+4)}$. We can decompose this into partial fractions:

Step 2: Setting up the equation

$$\frac{y+1}{(y+2)(y+4)} = \frac{A}{y+2} + \frac{B}{y+4}$$

Step 3: Solving for A and B

Multiplying both sides by $(y+2)(y+4)$, we get: $$y+1 = A(y+4) + B(y+2)$$ To find $A$, let $y = -2$: $$-2+1 = A(-2+4) + B(0) \implies -1 = 2A \implies A = -\frac{1}{2}$$ To find $B$, let $y = -4$: $$-4+1 = A(0) + B(-4+2) \implies -3 = -2B \implies B = \frac{3}{2}$$

Step 4: Substituting A and B

So, we have: $$\frac{y+1}{(y+2)(y+4)} = -\frac{1}{2(y+2)} + \frac{3}{2(y+4)}$$ Substituting back $y = x^2$, we get: $$\frac{x^2+1}{(x^2+2)(x^2+4)} = -\frac{1}{2(x^2+2)} + \frac{3}{2(x^2+4)}$$

Step 5: Integrating

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

Final Answer: $-\frac{1}{2\sqrt{2}} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right) + \frac{3}{4} \tan^{-1}\left(\frac{x}{2}\right) + C$

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concept of partial fractions and apply the standard integration formulas.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific procedure (partial fraction decomposition) and then use standard integration techniques.<\/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 and partial fractions, which are standard topics in the syllabus.