$$\cos 2x - \cos 2\alpha = (2\cos^2 x - 1) - (2\cos^2 \alpha - 1)$$$$\cos 2x - \cos 2\alpha = 2\cos^2 x - 2\cos^2 \alpha$$$$\cos 2x - \cos 2\alpha = 2(\cos^2 x - \cos^2 \alpha)$$
$$=2(\cos x - \cos \alpha)(\cos x + \cos \alpha)$$
Hence
$$\frac{\cos 2x-\cos 2\alpha}{\cos x-\cos \alpha} = \frac{2(\cos x - \cos \alpha)(\cos x + \cos \alpha)}{\cos x-\cos \alpha}$$
$$ = 2(\cos x + \cos \alpha)$$
Hence integral becomes:$$I = \int 2(\cos x + \cos \alpha) dx$$
$$I = \mathbf{2(\sin x + x \cos \alpha) + C}$$
AI Tutor Explanation
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Step-by-Step Solution
**Step 1: Apply Trigonometric Identity**
We know that \(\cos 2x = 2\cos^2 x - 1\). Therefore,
\(\cos 2x - \cos 2\alpha = (2\cos^2 x - 1) - (2\cos^2 \alpha - 1) = 2(\cos^2 x - \cos^2 \alpha)\).
**Step 2: Simplify the Integrand**
The integral becomes:
\(\int \frac{2(\cos^2 x - \cos^2 \alpha)}{\cos x - \cos \alpha} dx = 2\int \frac{(\cos x - \cos \alpha)(\cos x + \cos \alpha)}{\cos x - \cos \alpha} dx\)
\(= 2\int (\cos x + \cos \alpha) dx\)
**Step 3: Integrate**
\(2\int (\cos x + \cos \alpha) dx = 2\left[\int \cos x dx + \int \cos \alpha dx\right]\)
\(= 2[\sin x + x\cos \alpha] + C\)
\(= 2(\sin x + x\cos \alpha) + C\)
Correct Answer: 2(sin x + xcosα) + C
AI Suggestion: Option C
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because it requires the student to apply trigonometric identities and integration techniques to solve the problem.
Knowledge Dimension:PROCEDURAL
Justification:The question requires the student to follow a specific procedure involving trigonometric identities and integration rules to arrive at the solution.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of trigonometric identities and integration techniques as covered in the textbook.