Using Trignometric Integration

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In summary, Jon is having trouble with an integral. He is not sure how to solve it, and jpreed helps him by translating it into a formula that he can use in a graphing calculator.
  • #1
ntox101
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Homework Statement



I am having problems with this integral. I cannot find a similar example in the book(Calculus 8th Edition) or an odd numbered problem. Here it is

Integral [ 1 / csc [x] - 1 ] dx


Homework Equations



None to my knowledge.



The Attempt at a Solution



I have tried multiplying by the conjugate to form Integral [ csc[x] + 1 / cot^2 x ] dx . But for some reason I think I am doing something incorrect. I have also tried converting into terms of cosine and sine. No luck. :(
 
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  • #2
This is more of a trig problem than a calculus problem.

[tex]\int\frac{1}{\csc x -1}[/tex]

[tex]=\int\frac{1}{\frac{1}{\cos x}-1}[/tex]

[tex]=\int\frac{1}{\frac{1-\cos x}{\cos x}}[/tex]

[tex]=\int\frac{\cos x}{1 - \cos x}[/tex]

[tex]=\int\frac{(\cos x)(1 + \cos x)}{(1 - \cos x)(1 + \cos x)}[/tex]

[tex]=\int\frac{\cos x + \cos^2 x}{\sin^2 x}[/tex]

You should be able to do the rest. Hope this helps.
 
  • #3
ntox101 said:

Homework Statement



I am having problems with this integral. I cannot find a similar example in the book(Calculus 8th Edition) or an odd numbered problem. Here it is

Integral [ 1 / csc [x] - 1 ] dx
The way you have written the integrand is ambiguous. jpreed has made an assumption about what you meant, which might not be what the original problem is. Is the denominator of your integrand just csc(x) or is it csc(x) - 1? If it's the latter you must put parentheses around it to prevent ambiguity.

IOW, is this the integral?
[tex]\int (\frac{1}{csc(x)} - 1) dx[/tex]
(This is how most would interpret what you have written.)

Or is it this, which is how jpreed interpreted it?
[tex]\int \frac{1}{csc(x) - 1} dx[/tex]
ntox101 said:

Homework Equations



None to my knowledge.



The Attempt at a Solution



I have tried multiplying by the conjugate to form Integral [ csc[x] + 1 / cot^2 x ] dx . But for some reason I think I am doing something incorrect. I have also tried converting into terms of cosine and sine. No luck. :(
 
  • #4
Mark44,

jpreed's notation is correct according to this worksheet. I ended up splitting the integrals into two separate ones and using U-Substitution on the first set and just taking the integral of cot2x which is provided by an integration table in the back of my text.

Thanks for your help, both of you.

Jon
 
  • #5
This is a case of two wrongs making a right. You miswrote the integral that was in your worksheet, and jpreed translated it as what he thought you really meant.

My point is that if you write a rational expression such as a/b + c, you might mean this to be a/(b + c), but most would read this as (a/b) + (c).

Use parentheses!
 
  • #6
Yeah, I noticed that, hence when you try something similar in a graphing calculator/matlab/mathmatica it gives wrong answers. I will in the future, thanks.
 
  • #7
The graphing calculator/matlab/Mathematica are giving correct answers to what you entered, which doesn't happen to be the same as what you intended.
 
  • #8
To be clear, I realized the ambiguity in the integral as he had it written. If you take the time to read the entire original post he mentions multiplying the conjugate at the bottom.

I untangled the ambiguity from what he had written there and then worked out the rest.
 

1. What is trignometric integration?

Trignometric integration is a method used to solve integrals involving trigonometric functions, such as sine, cosine, and tangent.

2. Why is trignometric integration important?

Trignometric integration is important because it allows us to solve complex integrals involving trigonometric functions, which are commonly found in physics, engineering, and other scientific fields.

3. How do you use trignometric integration?

To use trignometric integration, you first need to identify the type of integral you have (sine, cosine, tangent, or a combination), then apply the appropriate trigonometric identities and integration rules to solve it.

4. What are the common mistakes to avoid when using trignometric integration?

Common mistakes to avoid when using trignometric integration include forgetting to use the chain rule, not simplifying expressions before integrating, and making sign errors when applying trigonometric identities.

5. Are there any tips for mastering trignometric integration?

Some tips for mastering trignometric integration include practicing with different types of integrals, memorizing common trigonometric identities, and always checking your work for mistakes. It also helps to have a good understanding of algebra and basic calculus concepts.

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