# How to find $\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10}\mathrm dx$

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.

• Express everything with $\cos(x)$ and $\sin(x)$. Then put $t=tg(\frac{x}{2})$. – pointer Aug 21 '14 at 12:09
• There might be a clever way with trig identities. But the only insight that immediately springs to mind is to replace $\cos nx$ with $\frac 12(e^{inx} + e^{-inx})$, then make the sub $u = e^{ix}$ and hope to god the algebra (factorisation of the denominator, partial fractions) is simple later on. – Deepak Aug 21 '14 at 12:09
• D'Moivers theorem can be used to find $\cos nx$ and $\sin nx$ in terms of $\cos$ and $\sin$ – Bumblebee Aug 21 '14 at 12:25

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.
• @Deepak. You are perfectly right ! I am sure that this is the way to build this kind of problem : start with the expansion of $\cos^nx$ as a function of multiple angles and ask the student to do the reverse ! Cheers :_) – Claude Leibovici Aug 21 '14 at 12:25