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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.

Rodrigo de Azevedo
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Stallmp
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1 Answers1

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

Dr. Sonnhard Graubner
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  • @Sonnhard, Could you please share your steps: My try: Write $x=2y$ $$1-2\cos6y=1-2(2\cos^23y-1)=-\dfrac{\cos9y}{\cos3y}$$

    $$\cos10y-\cos8y=-2\sin y\sin9y$$ Then how to reach at your expression. Please rectify me!

     – lab bhattacharjee Mar 04 '18 at 13:42
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    Please provide the entire solution though. – Stallmp Mar 04 '18 at 14:24
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    @Stallmp This is not a homework solving site. People will be more willing to help you if you show your own work. – A. Goodier Mar 04 '18 at 14:39
  • @rbird It's not homework. It's an integral that is beyond my knowledge, so how am I supposed to show my work? Asking questions should be easy, which is the motto of stackexchange. If you guys keep making up new rules it's extremely frustrating for the question askers, preventing them from asking anything at all. All I am asking is the steps for this integral. – Stallmp Mar 04 '18 at 14:41