 Hi and how are you all today? My name is Priyanka and I should be helping you with the following question. It says evaluate integral 0 to pi by 2 cos x upon 1 plus sin x into 2 plus sin x dx. So here let us proceed with the solution. Let us rewrite the given definite integral again. Now here we can use our substitution method to simplify this integral. Now here first of all let i be equal to integral cos x upon 1 plus sin x into 2 plus sin x dx. Now if we put sin x equal to t this implies cos x dx is equal to dt right. So we have i equal to dt upon 1 plus t into 2 plus t. Now we can solve it by using partial fraction and on doing so we can write integral dt upon 1 plus t into 2 plus t as integral 1 dt upon 1 plus t as the value of a we got as 1 minus 1 dt upon 2 plus t as the value of b after using partial fraction we got as minus 1. So we can write this as log mod 1 plus t minus log mod 2 plus t plus c which on substituting back the value we get as log 1 plus t the value of t is sin x minus log 2 plus sin x plus c which is log mod 1 plus sin x upon 2 plus sin x plus c. So we have integral 0 to pi by 2 cos x upon 1 plus sin x into 2 plus sin x dx equal to log mod 1 plus sin x upon 2 plus sin x 0 to pi by 2. Now we can find our answer that will be log mod 1 plus sin pi by 2 that is 1 upon 2 plus 1 minus log mod 1 plus 0 upon 2 plus 0. So we have log mod 2 upon 3 minus log 1 upon 2 which is equal to log 2 minus log 3 minus log 1 plus log 2. These two by combining can be written as log 4 minus log 3 as the value of log 1 we know is 0. So final answer is log mod 4 upon 3. So the answer to the question is log mod 4 upon 3 is the value of the given definite integral right. Hope you understood the procedure well and to take care of all your steps. Have a nice day ahead.