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∫(cos 8x – cos 7x)/ (1+ 2 cos 5x) ∫ (sin2x- sin2k)/ (sin x- sin k+ cosx- cosk)

  1. ∫(cos 8x – cos 7x)/ (1+ 2 cos 5x)
  2. ∫ (sin2x- sin2k)/ (sin x- sin k+ cosx- cosk)

Grade:12th pass

1 Answers

mycroft holmes
272 Points
7 years ago
\frac{\cos 8x - \cos 7 x}{1+2 \cos 5x} = \frac{-2 \sin \frac{15x}{2} \sin \frac{x}{2}}{3-4 \sin^2 \frac{5x}{2}}
 
We have \sin \frac{15x}{2} = \sin \left( 3 \times \frac{5x}{2}\right ) = \sin \frac{5x}{2} \left(3 - 4 \sin^2 \frac{5x}{2} \right )
 
Hence the expression simplifies to -2 \sin \frac{5x}{2} \sin \frac{x}{2} = \cos 3x - \cos 2x
which you can easily integrate.

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