I have a integral which seems difficult to me. Any help would be appreciated.
Find $$\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10}\mathrm dx$$
Also I wound like to know your thought process to solve integrals like these.
2026-03-30 17:05:22.1774890322
How to find $\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10}\mathrm dx$
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If you expand the cosines of multiple angles as functions of $\cos(x)$, you should arrive to $$\cos5x + 5\cos3x +10\cos x=16 \cos ^5 x$$ $$\cos6x+ 6\cos4x + 15\cos2x +10=32 \cos ^6 x$$ which means that $$\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10} \mathrm dx=\frac{1}{2}\int\sec x ~\mathrm dx$$ Now, use Weierstrass substitution.
I am sure that you can take from here.