Integrating csc(x): Easier Than You Think

  • Thread starter DrKareem
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In summary, the formula for solving the integral of cosecant (csc) is to use the substitution t=tan(x/2) and simplify the resulting expression. This can be done by separating the fractions and simplifying with a common denominator. Additionally, dividing by cos^2(x/2) in both the numerator and denominator of the left-hand side of the expression will also work.
  • #1
DrKareem
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haven't found a way of doing it so far. I have a feeling that it's extremely easy, and I'm missing how to do it somehow :/
 
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  • #2
do you mean cosec(x), the cosecans ? If so, just use the t=tan(x/2) formula's...

marlon
 
  • #3
I'm not sure what formula you're talking about.

[tex] \int \csc(x) dx [/tex]

If you take [tex]t=tan(\frac{x}{2})[/tex],

you'd get:

[tex]\frac {d}{dx} \tan(x)= \frac{1}{2}.sec^2(x)[/tex]

Not sure how to go from there...
 
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  • #4
What marlon meant, is the following:
[tex]csc(x)=\frac{1}{\sin(x)}=\frac{\cos^{2}(\frac{x}{2})+\sin^{2}(\frac{x}{2})}{2\sin(\frac{x}{2})\cos(\frac{x}{2})}=\frac{1+tan^{2}(\frac{x}{2})}{2tan(\frac{x}{2})}[/tex]

Substitute [tex]u=tan(\frac{x}{2})[/tex]
This implies:
[tex]\frac{du}{dx}=\frac{1}{2}\frac{1}{\cos^{2}(\frac{x}{2})}=\frac{1}{2}(u^{2}+1)[/tex]
Or:
[tex]dx=\frac{2du}{u^{2}+1}[/tex]
Hence, we have:
[tex]\int{csc(x)}dx=\int\frac{du}{u}=ln|u|+C=ln|tan(\frac{x}{2})|+C[/tex]
 
  • #5
I'm not sure how you did this equality:


[tex]\frac{\cos^{2}(\frac{x}{2 })+\sin^{2}(\frac{x}{2})}{2\sin(\frac{x}{2})\cos(\ frac{x}{2})}=\frac{1+tan^{2}(\frac{x}{2})}{2tan(\frac{x}{2})}[/tex]


Can you please clarify?

Other than that, it's all clear, thank you very much.
 
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  • #6
[itex]\int \csc x = \int \csc x \left(\frac{\csc x - \cot x}{\csc x - \cot x}\right) = \int \frac{du}{u} = \ln |csc x - cot x|[/itex]
(is that what we're talking about?)
 
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  • #7
yes, nice method :)

Still i would like someone to explain the question of my last post.
Thank you :)
 
  • #8
separate the fractions and simplify
[itex]\frac{\cos u}{2\sin u}+ \frac{\sin u}{2\cos u} =[/itex]
[itex]\frac{1}{2\tan u} + \frac{tan u}{2} [/itex]
get a common denominator and you're done.
 
  • #9
Still i would like someone to explain the question of my last post.

Just divide by cos^2(x/2) in both the numerator and the denominator of the LHS of the expression in question and it will drop straight out.
 
  • #10
Yes, excellent, so know i know i am stupid hehe :)

Thanks a lot for your help guys :)
 

What is "Integrating csc(x): Easier Than You Think"?

"Integrating csc(x): Easier Than You Think" is a mathematical concept that involves finding the integral of the cosecant function, which is commonly denoted as csc(x). It is a commonly used technique in calculus and can help solve many complex problems.

Why is integrating csc(x) considered easier than other techniques?

Integrating csc(x) is considered easier because it follows a specific formula and does not require a lot of complex algebraic manipulations. This makes it a more straightforward and efficient method for finding integrals.

What are the benefits of knowing how to integrate csc(x)?

Knowing how to integrate csc(x) can be beneficial in many ways. It can help solve complex mathematical problems, and also has real-world applications in fields such as physics and engineering. Additionally, understanding this concept can strengthen overall understanding of calculus and other mathematical concepts.

Are there any tips for easily integrating csc(x)?

One helpful tip for integrating csc(x) is to use the Pythagorean identity of sin^2(x) + cos^2(x) = 1. This can help simplify the integral and make it easier to solve. Additionally, practicing and familiarizing yourself with the basic formula can also make the process easier.

Are there any common mistakes to avoid when integrating csc(x)?

One common mistake to avoid when integrating csc(x) is forgetting to include the constant of integration. It is also important to carefully check any substitutions or algebraic manipulations made during the process to avoid calculation errors. Additionally, it is important to remember any special cases, such as when the argument of the csc(x) function is equal to 0 or pi.

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