Help with integration problem

[AjitZK]

Question:

My workings so far:

Solution:

I'm working on this question and I can't understand one part of the solution for 12c. In the attached image of the solution, I was wondering how the conclusion was made in the last line there since pi/2 came from (pi-x) for some reason. The solution attached is only the first part of the solution for 12c, as I wish to calculate the rest on my own. However, as I needed help to continue the question, I referred to the solution and didn't understand the next step. Thank you in advance and have a good day!

P.S. I checked my workings after I did them, so far no mistake

 

[Deleted member 4993]

Question:



My workings so far:


Solution:

I'm working on this question and I can't understand one part of the solution for 12c. In the attached image of the solution, I was wondering how the conclusion was made in the last line there since pi/2 came from (pi-x) for some reason. The solution attached is only the first part of the solution for 12c, as I wish to calculate the rest on my own. However, as I needed help to continue the question, I referred to the solution and didn't understand the next step. Thank you in advance and have a good day!

P.S. I checked my workings after I did them, so far no mistake

Let:

\(\displaystyle \displaystyle{\int_o^{\pi}\frac{xsin^3x}{1+cos^2x}dx \ = \ I}\)

then:

\(\displaystyle \displaystyle{\int_o^{\pi}\frac{(\pi - x)sin^3x}{1+cos^2x}dx \ = \ \pi*\int_o^{\pi}\frac{sin^3x}{1+cos^2x}dx \ - \ I}\)

So the line before the last line shows:

\(\displaystyle \displaystyle{I = \ \pi*\int_o^{\pi}\frac{sin^3x}{1+cos^2x}dx \ - \ I}\)

\(\displaystyle \displaystyle{2 * I = \ \pi*\int_o^{\pi}\frac{sin^3x}{1+cos^2x}dx}\)

and continue.......