Integrate Cos 2aDate: 08/15/2001 at 04:33:45 From: Katherine Watchorn Subject: Integration of cos 2a Dear Dr. Math, I was doing some integration and one of the problems was to integrate cos2a. The answer given in the book is sin 2a/2. The textbooks that I have tell me that integrating cos a gives me sin a, and tell me in detail how that was arrived at. But I have searched everywhere for information on cos2a and how the result sin 2a/2 is arrived at. Does this follow a pattern, i.e. cos 3a = sin 3a/3 cos 4a = sin 4a/4 or cos na = sin na/n ? Many thanks for your help with previous questions; it was much appreciated. Katherine Watchorn Date: 08/15/2001 at 12:59:31 From: Doctor Peterson Subject: Re: Integration of cos 2a Hi, Katherine. In general, the formula is [INT] cos(ax)dx = sin(ax) / a You can prove this by taking the derivative of the right side: d/dx[1/a sin(ax)] = 1/a cos(ax) d/dx(ax) = 1/a cos(ax) * a = cos(ax) This required only a simple application of the chain rule. Not knowing this more general formula, you can obtain it by substitution. If we let u = ax, then du = a dx and dx = du/a, so [INT]cos(ax)dx = [INT]cos(u) du/a = sin(u)/a = sin(ax)/a Here I first replaced ax with u and dx with du/a, then integrated, and finally replaced u with ax again. The same method is useful everywhere, so you should learn it well and even be able to do it in your head. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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