Let \[ I = \int {\frac{{\cos 5x + \cos 4x}}{{1 – 2\cos 3x}}} dx = \int {\frac{{2\cos \frac{{9x}}{2} \cdot \cos \frac{x}{2}}}{{1 – 2\left( {2{{\cos }^2}\frac{{3x}}{2} – 1} \right)}}} dx\]


and \[\left. {\cos 2x = 2{{\cos }^2}x – 1} \right]\]

therefore,\[ I = \int {\frac{{2\cos \frac{{9x}}{2} \cdot \cos \frac{x}{2}}}{{3 – 4{{\cos }^2}\frac{{3x}}{2}}}} dx = – \int {\frac{{2\cos \frac{{9x}}{2} \cdot \cos \frac{x}{2}}}{{4{{\cos }^2}\frac{{3x}}{2} – 3}}} dx\]

therefore,\[ I = \int {\frac{{2\cos \frac{{9x}}{2} \cdot \cos \frac{x}{2}}}{{3 – 4{{\cos }^2}\frac{{3x}}{2}}}} dx = – \int {\frac{{2\cos \frac{{9x}}{2} \cdot \cos \frac{x}{2}}}{{4{{\cos }^2}\frac{{3x}}{2} – 3}}} dx\]

\[ = – \int {\frac{{2\cos \frac{{9x}}{2} \cdot \cos \frac{x}{2} \cdot \cos \frac{{3x}}{2}}}{{4{{\cos }^3}\frac{{3x}}{2} – 3\cos \frac{{3x}}{2}}}} dx\quad \left[ {} \right.\] multiplying and dividing by \[ \left. {\cos \frac{{3x}}{2}} \right] \]

\[ = – \int {\frac{{2\cos \frac{{9x}}{2} \cdot \cos \frac{x}{2} \cdot \cos \frac{{3x}}{2}}}{{\cos 3 \cdot \frac{{3x}}{2}}}} dx = – \int 2 \cos \frac{{3x}}{2} \cdot \cos \frac{x}{2}dx\]

\[ = – \int {\left\{ {\cos \left( {\frac{{3x}}{2} + \frac{x}{2}} \right) + \cos \left( {\frac{{3x}}{2} – \frac{x}{2}} \right)} \right\}} dx\]

\[ = – \int {(\cos 2x + \cos x)} dx\]

\[ = – \left[ {\frac{{\sin 2x}}{2} + \sin x} \right] + C\]

\[ = – \frac{1}{2}\sin 2x – \sin x + C\]

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