Integrating a Quadruple-Angle Trigonometric Expression or TwoDate: 01/13/2011 at 06:49:06 From: abhishek Subject: integrate the (1+cos4x)/(cosx-tanx) I am having a problem integrating this: (1 + cos4x)/(cosx - tanx) Date: 01/13/2011 at 12:28:43 From: Doctor Ali Subject: Re: integrate the (1+cos4x)/(cosx-tanx) Hi Abhishek! Thanks for writing to Dr. Math. Can you show me what steps you've tried already? That would help me tailor a response to your problem. If you don't have any idea how to begin, here is where you can start: - | 1 + cos(4x) | ----------------- dx | cos(x) - tan(x) - Note that 1 + cos(4x) = 2 cos^2(2x) tan(x) = sin(x)/cos(x) So, we can write the integral as - | 2 cos^2(2x) | ------------------------ dx | cos(x) - sin(x)/cos(x) - Multiply the top and bottom by cos(x) and cos(2x) in terms of sin(x). This gives - | 2 (1 - 2 sin^2(x))^2 cos(x) | ----------------------------- dx | cos^2(x) - sin(x) - Recognizing another trigonometry identity in the denominator, re-write the integrand as - | 2 (1 - 2 sin^2(x))^2 cos(x) | ----------------------------- dx | 1 - sin^2(x) - sin(x) - Now, consider u = sin(x) You'll have du = cos(x) dx With this substitution, you can write the integral as - | 2(1 - 2u^2)^2 | --------------- du | 1 - u - u^2 - Can you take it from here? Please write back if you still have any difficulties. - Doctor Ali, The Math Forum http://mathforum.org/dr.math/ Date: 01/13/2011 at 12:46:08 From: abhishek Subject: integrate the (1+cos4x)/(cosx-tanx) I meant to ask you for help integrating (1 + cos4x)/(cotx - tanx) Note the denominator has a cotangent term, rather than cosine. Thanks. Abhishek Network Engineer INDIA Date: 01/13/2011 at 12:59:15 From: Doctor Ali Subject: Re: integrate the (1+cos4x)/(cosx-tanx) Hi Abhishek! Thanks for writing to Dr. Math. This slightly different trigonometric expression is even easier to integrate; just write tan(x) and cot(x) in the bottom in terms of sin(x) and cos(x). Please be informed that Ask Dr. Math is a place for people who are interested in math to help each other learn more about it. While we're happy to help you with a question from your homework (or job!), we won't give you the answer. It's your responsibility to solve the problem. Our goal is to help you find out why you're stuck, so you can get un-stuck. Therefore, one of the best ways to get our help is to show that you've made a serious effort to answer your question on your own. The best way to do that is to show us your work (even if you're sure it's wrong), or at least tell us what you've been thinking. If you submit your questions without comment and then sit back and wait for the answers, you may find yourself waiting a long time! Please write back if you still have any difficulties. - Doctor Ali, The Math Forum http://mathforum.org/dr.math/ |
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