Integration Problem: Is My Book Wrong?

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In summary, the conversation is about an integration problem involving the functions tangent and secant. The problem was solved using different methods, including substitution and using trigonometric identities. There was a discussion about the notation used and the different approaches to solving the problem. Ultimately, it was determined that the solutions were equivalent and the conversation ended on a humorous note.
  • #1
mad
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I have this integration problem that I did but it doesn't give me the right answer. But there are like 3 other similar exercices I did the same way and I got all the right answers.. maybe it is my book (I don't think so ... :p)

It's an integration problem:

[tex]\int tg^3(4x) sec^4(4x)dx[/tex]

heres what I did:

[tex]\int (sec^2(4x)-1)(tg(4x))(sec^4(4x))dx[/tex]
= [tex]\int (sec^6(4x) - sec^4(4x)) (tg4x) dx [/tex]

u= sec 4x
du = 4(sec4x)(tg4x)dx --> dx = du/(4(sec4x)(tg4x))

=[tex]\int \frac{(u^6 - u^4) du}{4u} [/tex]

(replaced the sec(4x) at denom. with u since u=sec4x)

=[tex]\frac{1}{4}\int u^5 - [/tex] [tex]\frac{1}{4}\int u^3 [/tex]
= [tex]\frac{1}{24}sec^6(4x) -[/tex] [tex]\frac{1}{16} sec^4(4x) [/tex] +C


Add anything you want! Thanks everyone

BTW the answer in my book is
(1/16) tg^4 (4x) + (1/24) tg^6(4x)
I tried it in my calc with an x and it doesn't give the same answer.
 
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  • #2
Actually, those aer the same answers (up to that constant, if you put it directly into your calculator, they differ already by a constant that doesn't matter in this type of integration...add 1/48 to your answer to get the other). They just went another direction in the integration.
 
  • #3
What the hell is tg(x)?
 
  • #4
Here's another way.Denote the integral by I and write everything in terms of sine a cosine...

[tex] I=\int \left(\frac{\sin^{3}4x}{\cos^{7}4x}\right) \ dx [/tex]

[tex] 4x=u \Rightarrow 4dx=du [/tex]

[tex] I=\frac{1}{4}\int \left(\frac{\sin^{4}u}{\cos^{7}u}\right) \ du
=-\frac{1}{4}\int \left(\frac{1-\cos^{2}u}{\cos^{7}u}\right) \ d(\cos u) =-\frac{1}{4}\left[\frac{(\cos u)^{-6}}{-6}-\frac{(\cos u)^{-4}}{-4}\right] +\mathcal{C}[/tex]

Therefore,reversing the substitution made

[tex]I=\frac{1}{24}\frac{1}{\cos^{6}4x}-\frac{1}{16}\frac{1}{\cos^{4}4x}+\mathcal{C} [/tex]

Daniel.
 
  • #5
whozum said:
What the hell is tg(x)?

Tangent of (x),what else? :rolleyes:

Daniel.
 
  • #6
dextercioby said:
Here's another way.Denote the integral by I and write everything in terms of sine a cosine...

[tex] I=\int \left(\frac{\sin^{3}4x}{\cos^{7}4x}\right) \ dx [/tex]

[tex] 4x=u \Rightarrow 4dx=du [/tex]

[tex] I=\frac{1}{4}\int \left(\frac{\sin^{4}u}{\cos^{7}u}\right) \ du
=-\frac{1}{4}\int \left(\frac{1-\cos^{2}u}{\cos^{7}u}\right) \ d(\cos u) =-\frac{1}{4}\left[\frac{(\cos u)^{-6}}{-6}-\frac{(\cos u)^{-4}}{-4}\right] +\mathcal{C}[/tex]

Therefore,reversing the substitution made

[tex]I=\frac{1}{24}\frac{1}{\cos^{6}4x}-\frac{1}{16}\frac{1}{\cos^{4}4x}+\mathcal{C} [/tex]

Daniel.


Thanks for your help, Daniel.
I see my solution was okay. We had to do the problem by exponents of sec and tg. (sorry I don't know what the method is called in english)

However, what does d(cos u) means in your solution? and I know you replaced a sin^4 (u) by (1-cos^2 (u)), but where is the other sin^2 (x)
Surely it is the d(cos u) you used, but I'm not familiar with this notation
 
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  • #7
dextercioby said:
Tangent of (x),what else? :rolleyes:

Daniel.

What happened to tan(x)?
 
  • #8
whozum said:
What happened to tan(x)?

Skin cancer you know...the poor devil.
 

Related to Integration Problem: Is My Book Wrong?

1. Why is integration important in science?

Integration is important in science because it allows us to combine different pieces of information or data to form a more complete understanding of a particular phenomenon or system. It helps us to connect different concepts and theories and see how they relate to each other.

2. How do I know if my book is wrong about integration?

If you are unsure about the accuracy of your book's information on integration, you can cross-reference it with other reliable sources such as scientific journals, textbooks, or consult with a knowledgeable expert in the field. It is important to critically evaluate information and not rely on a single source for accuracy.

3. Can integration be used in all scientific fields?

Yes, integration can be applied in all scientific fields as it is a fundamental concept in problem-solving and understanding complex systems. It is commonly used in physics, mathematics, biology, chemistry, and many other disciplines.

4. What are some common mistakes made in integration?

One of the most common mistakes in integration is not being familiar with the basic rules and techniques, resulting in incorrect calculations. Another mistake is not paying attention to the limits of integration or forgetting to include them altogether. It is also important to be careful with signs and not to mix up positive and negative values.

5. How can I improve my integration skills?

The best way to improve your integration skills is to practice regularly and familiarize yourself with the different techniques and rules. You can also seek help from a tutor or join a study group to work on integration problems together. Additionally, understanding the underlying concepts and theories behind integration can also help improve your skills in this area.

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