Your approach is absolutely correct.
first step : Integral sin^2x cos^2x dx = 1/4 Integral sin^2 ( 2x ) dx
Now use the identity cos 2x = 1 - sin^2 x
For some reason I am having trouble with the integral: sin^{2}xcos^{2}x dx
I have tried a few identities with no luck.
1st try
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(1/2) | (2sinxcosx)^{2} dx
2nd try
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(1/4) | (1 - cos(2x))(1 + cos(2x)) dx
The website I am looking at keeps saying to use integration by parts, which makes no sense to me.
Thanks for your help.
Hello,
You can use the double angle formula:
Thus
Now write and subsequently . So, .
So, the integral is now: .
Integrate that and you get . The only identities you need are and the double angle formulae , and . No integration by parts is necessary.
I think you're along the right lines for one possible solution with your first try.
Using the identity:
we can say:
now, in my understanding we can't integrate , so:
therefore the integral now becomes:
This link gives some quick advice about how to proceed with combined sine/cosine integrals. It doesn't actually explain why these rules of thumb work, but if you do several examples I think you'll figure out the why's by yourself. (Or we can help, too!)
-Dan