 Hi and how are you all today? The question says evaluate i is equal to integral 0 to pi x sin square x upon 1 plus cos square x dx. Now let's proceed with the solution. We have the given integral as 0 to pi x sin square x upon 1 plus cos square x dx. Now here on subtracting sorry we have we know that sin pi minus x is equal to sin x right. So we have i equal to integral 0 to pi pi minus x sin pi minus x upon 1 plus cos square pi minus x dx. So as we know this we have or i equal to 0 to pi x minus sorry pi minus x sin pi minus x is sin x and cos pi minus x is also cos x. So we have 1 plus cos x over here. Now this be first equation and let this be the second equation. So on adding the first equation and the second equation we get 2i is equal to integral 0 to pi pi sin x minus x sin x plus x sin x upon 1 plus cos square x dx. Now here these 2 will get cancelled out. We have 0 to pi pi which is a constant which will come out of the integral sin sin x upon 1 plus cos square x dx. Now if we put cos x equal to t then minus 2 sin x dx will be equal to dt and our limits will also change from 0 to pi to become minus 1 to 1 dt over 1 plus t square. So we have 2i equal to pi tan inverse t back it close minus 1 to 1 on applying the limits we have pi tan inverse 1 is pi by 4 minus tan inverse of minus 1 is minus pi by 4. So we have i equal to pi into pi by 4 plus pi by 4 is 2 pi by 4. So we have i equal to pi square by 2 and this is the required answer to the given question. Hope you understood it well and enjoy it too. Have a nice day.