JAMIA 2021 — Mathematics PYQ

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Question

JAMIA 2021 — Mathematics PYQ

JAMIA | Mathematics | 2021

$\int \frac{c o s 2 x - c o s 2 θ}{c o s x - c o s θ} d x$ equal to

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Explanation

1. Using the identity $\cos 2A = 2\cos^2 A - 1$ :

\int \frac{(2\cos^2 x - 1) - (2\cos^2 \theta - 1)}{\cos x - \cos \theta} dx

2. Simplify the numerator:

\int \frac{2\cos^2 x - 1 - 2\cos^2 \theta + 1}{\cos x - \cos \theta} dx

\int \frac{2\cos^2 x - 2\cos^2 \theta}{\cos x - \cos \theta} dx

3. Factor out 2 and use $a^2 - b^2 = (a-b)(a+b)$ :

2 \int \frac{\cos^2 x - \cos^2 \theta}{\cos x - \cos \theta} dx

2 \int \frac{(\cos x - \cos \theta)(\cos x + \cos \theta)}{\cos x - \cos \theta} dx

4. Cancel the common term $(\cos x - \cos \theta)$ :

2 \int (\cos x + \cos \theta) dx

5. Integrate with respect to $x$ (Note: $\cos \theta$ is a constant):

2 \left[ \int \cos x dx + \int \cos \theta dx \right]

2 [\sin x + x\cos \theta] + C

Final Answer:

2(\sin x + x\cos \theta) + C

Choose the correct answer:

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