Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Integrating a Quadruple-Angle Trigonometric Expression or Two

Date: 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/ 
Associated Topics:
High School Calculus
High School Trigonometry

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/