# What is the integral of #cos^6(x)#?

See explanation.

This will be a long answer.

So what you want to find is:

There's a rule of thumb that you can remember: whenever you need to integrate an even power of the cosine function, you need to use the identity:

First we split up the cosines:

Now you could apply FOIL twice, but I would rather use Newton's Binomial theorem. Following from this theorem is that

Let's apply this to the integral.

Now we can already splice this integral up a bit:

Whenever you have an odd power of cosines, you can do the following:

So

Now we have all our parts to complete the integral. Remember that we had:

You could simplify this a bit, which isn't that hard, I'll leave that as a challenge to you :D.

I hope this helps. It was fun!

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I would like to present an alternative approach - based on the idea that "complex" is simple!

We know that

and that

Using these we get

We now use standard integrals to evaluate :

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The integral of cos^6(x) dx equals (1/6)(cos^5(x) * sin(x) + (5/6)(∫cos^4(x) dx).

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