anti-derivative problem

[juzt10]

Hi.
Can anyone explain how [1 dx / ((csc(2x))^2 - cot(2x))] integrated?
I used online calculators and they showed different models of answer.

I prefer this answer: http://www.integral-calculator.com/#expr=1/((csc(2x))^2-cot(2x))
Rather than: http://www.wolframalpha.com/input/?i=integral+1%2F%28%28csc%282x%29%29^2-cot%282x%29%29

I know that: int du/u = ln |u|, but idk how to make that ln|tan^2 (2x)| and how the arctan come from.
Any help is highly appreciated.
Thanks.

 

[Deleted member 4993]

Hi.
Can anyone explain how [1 dx / ((csc(2x))^2 - cot(2x))] integrated?
I used online calculators and they showed different models of answer.

I prefer this answer: http://www.integral-calculator.com/#expr=1/((csc(2x))^2-cot(2x))
Rather than: http://www.wolframalpha.com/input/?i=integral+1%2F%28%28csc%282x%29%29^2-cot%282x%29%29

I know that: int du/u = ln |u|, but idk how to make that ln|tan^2 (2x)| and how the arctan come from.
Any help is highly appreciated.
Thanks.

You know from trigonometrical identities and quadratic factorization:

csc²(2x) - cot(2x) = cot²(2x) - 1 - cot(2x) = [cot(2x) - σ₁][cot(2x) - σ₂]
where

\(\displaystyle \displaystyle{\sigma_{1,2} \ = \ \dfrac{1 \pm \sqrt{5}}{2}}\)..............edited

Now use partial fractions and integrate.

 

[cindy8559]

Hi.
Can anyone explain how [1 dx / ((csc(2x))^2 - cot(2x))] integrated?
I used online calculators and they showed different models of answer.

I prefer this answer: https://gpa-calculator.online /#expr=1/((csc(2x))^2-cot(2x))
Rather than: http://www.wolframalpha.com/input/?i=integral+1/((csc(2x))^2-cot(2x))

I know that: int du/u = ln |u|, but idk how to make that ln|tan^2 (2x)| and how the arctan come from.
Any help is highly appreciated.
Thanks.

I used it up here and it feels good