What is $\int \frac{\cos(5x)-\cos(4x)}{1-2 \cos(3x)} \, \mathrm d x$?

Question

I came across this question on a YouTube video and I have no idea how to solve this. I know the following strategies (most important ones):

• Integration by parts
• Substitution
• Trigonometric substitution
• Integration by partial fractions
• Power rule
• Complex numbers
• Summation and limits

Is it possible to solve this integral with these techniques? If not, which technique would you use? I am interested in one of the EASIEST ways to solve this integral. I am also interested in the full solution.

Answer 1

hint: your Integrand simplifies to $$\frac{2\sin^2(\frac{x}{2})(2\cos(x)+1)(2\cos(3x)-1)}{2\cos(3x)-1}$$