Integrate cos^22x
To integrate cos^22x, also written as ∫cos22x dx, cos squared 2x, (cos2x)^2, and cos^2(2x), we start by considering standard trig identities to simplify the integral.
We recall the double angle formulae and rearrange it for cos2x.
We recall the Pythagorean identity and rearrange it for sin2x.
We substitute the expression for sin2x into the double angle formulae. We then group the cos2x terms on the LHS and simplify to give the above expression. As you can see, the LHS is looking more like our integration problem except that the angle we require is 2x. To get 2x we use a little trick as shown below.
If we multiply the angles on both sides by 2, then we have not changed anything and the equation remains balanced, however, now the LHS looks exactly like our integration problem.
As you can see, our integration problem is now in a different form but means the same thing, for the reasons explained in the steps before. The first term is simple to integrate, but the second term will need more thinking.
We let u = 4x, and therefore du/dx = 4.
We rearrange to give an expression for dx.
Therefore, as you can see, we now have a new integration in terms of u.
We end up with a simple trig integration.
Hence, we now have a simple answer, and we also replace u with 4x.
Finally, we are able to solve our original integration problem by substituting the answer for the second term, and C is the integration constant.