 Hi and welcome to the session. Let's work out the following question. The question says evaluate integral cos 2x divide into log sin x dx where limit goes from pi by 4 to pi by 2 So let us see the solution to this question Let us call it i. So i is equal to integral cos 2x into log sin x dx where the limit goes from pi by 4 to pi by 2 Now integrating by parts We get We see that this becomes the first function. This will be the second function. So integrating by parts we will have first function into integral of second function minus integral of differentiation of first function into integral of second function the limit goes from pi by 4 to pi by 2 This is equal to log sin x into sin 2x divided by 2 minus integral Now dy dx of log sin x is 1 upon sin x into cos x into sin 2x divided by 2 dx limit goes from pi by 4 to pi by 2 This is equal to log sin pi by 2 into sin pi by 2 minus log of sin pi by 4 into sin pi by 2 the whole divided by 2 minus integral. Now sin 2x can be written as 2 sin x cos x into cos x divided by 2 sin x dx where limit goes from pi by 4 to pi by 2 Since sin pi by 2 is 0 so this in Sorry since Sin pi by 2 is 1 so we will have log 1 log 1 is 0 also sin pi is 0 So this term will become 0 and we will have 0 minus Sin pi by 2 is 1 so 1 by 2 into log Sin pi by 4 is 1 upon root 2 Minus integral where limit goes from pi by 4 to pi by 2 2 sin x gets cancelled from numerator denominator We have cos square x and cos square x can be written as 1 plus cos 2x divided by 2 dx This is equal to minus 1 by 2 log 1 by root 2 minus x by 2 where limit goes from pi by 4 to pi by 2 minus Sin 2x divided by 4 where limit goes from pi by 4 to pi by 2 This is equal to minus 1 by 2 log 1 by root 2 Minus now if we put x to be equal to pi by 2 we get pi by 4 Minus now we put x to be equal to pi by 4 so pi by 8 Minus now we put x to be pi by 2 we have sin pi by 4 Minus Sin this will be pi by 2 by 4 This is equal to minus 1 by 2 log 1 by root 2 Minus pi by 8 Minus sin pi is 0. So we have minus Minus 1 by 4 because sin pi by 2 is 1 This is equal to minus 1 by 2 log 1 by root 2 minus pi by 8 plus 1 by 4 So this is our answer to this question. I hope that you understood the solution I enjoyed the session. Have a good day