00:01
So function given is f of xy which is tan inverse y times i plus minus xy square upon 1 plus y squared j.
00:13
So this is function of xy and r is given as since this is a curve.
00:17
So r is given as 2 cost t i plus 2 sine t j.
00:24
So our d r will be on differentiating this we get minus 2 sine t i plus 2 .000 t i plus 2.
00:30
T j d t correct and it is from 0 to 2 pi therefore our function curve will be from 0 to 2 pi that is our c and f in terms of t t can be written as tan inverse of y which is 2 sine t i plus here we have minus x which is 2 cos t then we have y square which is 2 sine t square over 1 plus 2 sine t whole square j dot d r and d r is what minus 2 sine t i plus 2 cos t j d t great so now if you multiply this we know i times i is 1 and j times j is 1 and press other at 0 so this will become 0 to 2 pi tan inverse of 2 s t 2 t times 2 sine t in negatives.
01:26
So i'll write minus sign here.
01:28
And this plus minus is minus 2 cost.
01:31
This two costs and these two cause will become.
01:34
So 2 times 2 cost square t and this is 4 sine square t over 1 plus 4 sine square t.
01:43
Great.
01:45
So now we have our function and we just need to integrate it.
01:47
So we'll apply, integrate this part first and this one separately.
01:52
Right so let me separate this integration so this is 0 to 2 pi minus 2 sine t i'll write this term first times tan inverse of 2 sine t d t minus this two twos are 4 and 4 times 4 is 16 so 16 cos square t sine square t over 1 plus 4 sine square t d t great and this is limit is 0 to 2 pi so now here one is tan inverse and one is signs so we'll apply u into v and since inverse is not integrable directly so we'll take these as an integration function this one is a differentiation function okay so we know u into v rule right and if you don't remember this is u times v d x then answer is u times v d x integration of v minus integration differentiation of what we have not integrated times the same integration to the whole integration right so now if we see here and here we have to remember that v is the function which can be integrated and u is function which differentiates either it terminates after multiple differentiation or it can be differentiated.
02:56
So here tan inverse we don't have direct integration so that we cannot take as v.
03:01
So we'll take it as u.
03:02
So applying that formula we get tan inverse 2 sine t integration of 2 sine t 0 to and here remember that the limit is applied the entire function minus integration of 0 to 2 pi.
03:18
Now we have to differentiate this tan inverse term.
03:20
So differentiation of tan inverse is 1 over 1 plus x square, if function is x.
03:26
So 2 sine t square times differentiation of what is our argument.
03:31
So that is 2 and differentiation of sine is cos t.
03:34
And here integration of 2 sine t again.
03:39
2 sine t d t and then again we have to integrate you this entire term.
03:43
And this term will remain as it is.
03:45
So let me write as i1...