Integration HintsDate: 02/23/99 at 19:08:03 From: Scott Subject: Question on Integration Can you help me integrate: (cos[x])^4 I have spent a full hour and I keep getting stuck, at a different point each time, with a different answer. I think you have to split it up and use identities instead, but I do not know which ones to use. If you could help me out with this, by writing me either the work, or a step-by-step proccess of how to solve this, I would GREATLY appreciate it. Date: 02/23/99 at 19:37:16 From: Doctor Wilkinson Subject: Re: Question on Integration I am not going to do all the work, but I can give you some useful hints. There are a lot of ways to go about this, but one of the simplest is to start by using the identity sin^2 x + cos^2 x = 1, and rewriting the integral as the integral of cos^2 x cos^2 x dx = cos^2 x (1 - sin^2 x) dx, which is the integral of cos^2 x dx - the integral of sin^2 x cos^2 x dx. Next I would use the double-angle formulae cos(2x ) = cos^2 x - sin^2 x = 2 cos^2 x - 1 to express the first integral in terms of the integral of cos (2x) dx and sin(2x) = 2 sinx cos x to express the second integral in terms of the integral of sin^2(2x), which again can be expressed in terms of the integral of cos(2x) dx by using the same double-angle formula for the cosine in the form cos(2x) = 1 - 2 sin^2 x. I hope this helps. You have to proceed carefully and watch the signs and so on. - Doctor Wilkinson, The Math Forum http://mathforum.org/dr.math/ |
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