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### 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

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

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
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

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