 Hello and welcome to the session. In this session we discussed the following question that says evaluate integral 0 to pi by 2 to log sin x minus log sin 2x dx. Before we move on to the solution let's recall one property of definite integrals. We have integral 0 to a, fx dx is equal to integral 0 to a, f of a minus x dx. This is the key idea that we use for this question. Let's move on to the solution now. We take let i be equal to the given integral 0 to pi by 2 to log sin x minus log sin 2x dx. So this is equal to integral 0 to pi by 2 to log sin x minus log. Now we know that sin 2x is equal to 2 sin x into cos x. So we have here log 2 sin x into cos x dx. So further we get this is equal to integral 0 to pi by 2 to log sin x minus log 2 plus log sin x plus log cos x dx. And so this is further equal to integral 0 to pi by 2 to log sin x minus log 2 minus log sin x minus log cos x dx. So this is equal to integral 0 to pi by 2 log 2 log sin x minus log sin x is log sin x dx minus integral 0 to pi by 2 log 2 dx minus integral 0 to pi by 2 log cos x dx. This is i. Now applying the property given in the key idea we get this is equal to integral 0 to pi by 2 log sin of pi by 2 minus x dx minus log 2 integral 0 to pi by 2 dx minus integral 0 to pi by 2 log cos x dx. Now this is further equal to integral 0 to pi by 2 log. Now sin pi by 2 minus x is cos x. So here it would be log cos x dx minus log 2 into x then it goes from 0 to pi by 2 minus integral 0 to pi by 2 log cos x dx. Now this term cancels with this term and this is equal to minus log 2 into pi by 2 minus 0 this is i. So further i is equal to minus pi by 2 into log 2. Thus we get the value of the given integral integral 0 to pi by 2 2 log sin x minus log sin 2x dx is equal to minus pi by 2 into log 2. So this is our final answer. This completes the session. Hope you have understood the solution of this question.