cos-7x-cos-8x-1-2cos-5x-dx-

Question Number 167596 by cortano1 last updated on 20/Mar/22

\int \frac{\cos 7 x - \cos 8 x}{1 + 2 \cos 5 x} dx = ?

Answered by MJS_new last updated on 20/Mar/22

\frac{\cos 7 x - \cos 8 x}{1 + 2 \cos 5 x} =

[c = \cos x]

= - \frac{(c - 1) (2 c + 1) (4 c^{2} + 2 c - 1) (16 c^{4} - 8 c^{3} - 16 c^{2} + 8 c + 1)}{(2 c + 1) (16 c^{4} - 8 c^{3} - 16 c^{2} + 8 c + 1)} =

= - 4 c^{3} + 2 c^{2} + 3 c - 1 = \cos 2 x - \cos 3 x

\Rightarrow

\int \frac{\cos 7 x - \cos 8 x}{1 + 2 \cos 5 x} d x = \int (\cos 2 x - \cos 3 x) d x =

= \frac{\sin 2 x}{2} - \frac{\sin 3 x}{3} + C

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