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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.3
Find the integrals of the functions in Exercises 62 to 83.
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Question : 74 of 271
Marks: +1, -0
cos⁡2x−cos⁡2αcos⁡x−cos⁡α\frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}
Solution:  
Let I=∫cos⁡2x−cos⁡2αcos⁡x−cos⁡αdxI = \int \frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha} dx
=∫(2cos⁡2x−1)−(2cos⁡2α−1)cos⁡x−cos⁡αdx= \int \frac{(2\cos^2 x - 1) - (2\cos^2 \alpha - 1)}{\cos x - \cos \alpha} dx
=∫2(cos⁡2x−cos⁡2α)cos⁡x−cos⁡αdx= \int \frac{2(\cos^2 x - \cos^2 \alpha)}{\cos x - \cos \alpha} dx
=∫2(cos⁡x−cos⁡α)(cos⁡x+cos⁡α)cos⁡x−cos⁡αdx= \int \frac{2(\cos x - \cos \alpha)(\cos x + \cos \alpha)}{\cos x - \cos \alpha} dx
=2∫(cos⁡x+cos⁡α)dx= 2 \int (\cos x + \cos \alpha) dx
=2∫cos⁡x dx+∫2cos⁡α dx= 2 \int \cos x \, dx + \int 2 \cos \alpha \, dx
=2sin⁡x+2xcos⁡α+C= 2 \sin x + 2x \cos \alpha + C
=2(sin⁡x+xcos⁡α)+C= 2 (\sin x + x \cos \alpha) + C
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