 Hello and welcome to the session. Let us discuss the following question which says evaluate integral of x sin x upon 1 plus cos square x dx from 0 to pi. Before proceeding for the solution let's recall few formulas which will be helpful in solving this question. So first of all we have integral of f of x dx from 0 to a is equal to integral of f of a minus x dx from 0 to a. Second is integral of f of x dx from a to b is equal to minus integral of f of x dx from b to a. Third we have integral of 1 upon 1 plus x square dx is equal to tan inverse x. And lastly tan inverse of minus x is equal to minus tan inverse of x. So this is the key idea for this question. Now let's move on to its solution. We need to integrate x into sin x upon 1 plus cos square x dx from 0 to pi. So let us assume this to be equal to i. Now let us use the first property. So this will be equal to integral of pi minus x into sin of pi minus x upon 1 plus cos square pi minus x dx from 0 to pi. Because integral of f of x dx from 0 to a is equal to integral of f of a minus x dx from 0 to a. Now we know that sin pi minus x is equal to sin x and cos pi minus x is equal to minus cos x. So this implies that cos square pi minus x will be equal to square of minus cos x that is cos square x. Now let us substitute the values of sin pi minus x and cos pi minus x over here. Thus i will be equal to integral of pi minus x into sin x upon 1 plus cos square x dx from 0 to pi. So this will be equal to integral of pi into sin x upon 1 plus cos square x dx from 0 to pi minus integral of x into sin x upon 1 plus cos square x dx from 0 to pi. Now in the beginning we assumed integral of x into sin x upon 1 plus cos square x dx from 0 to pi equal to i. Thus this will be equal to integral of pi into sin x upon 1 plus cos square x dx from 0 to pi minus i. So this implies i plus i is equal to integral of pi into sin x upon 1 plus cos square x dx from 0 to pi that is 2i is equal to pi into integral of sin x upon 1 plus cos square x dx from 0 to pi. Now to integrate this let us substitute cos x as t. So let t is equal to cos x. This implies dt by dx is equal to minus sin x or minus dt is equal to sin x dx. So when x is equal to 0 then t will be equal to cos 0 which is equal to 1 and when x is equal to pi then t will be equal to cos pi which is equal to minus 1 thus 2i will be equal to pi into integral of minus dt upon 1 plus t square from 1 to minus 1. Now let us use this property. So this is equal to pi into integral of minus 1 to 1 dt upon 1 plus t square because integral of f of x dx from a to b is equal to minus into integral of f of x dx from b to a and we already know that integration of 1 upon 1 plus x square dx is equal to tan inverse x. So integration of 1 upon 1 plus t square dt will be equal to tan inverse t. So this implies that 2i is equal to pi into tan inverse of t from minus 1 to 1 which will be equal to pi into tan inverse of 1 minus tan inverse of minus 1 that is pi into tan inverse of 1 plus tan inverse of 1 because tan inverse of minus x is equal to minus tan inverse of x. So this gives that 2i is equal to pi into pi by 4 plus pi by 4 because tan inverse of 1 is equal to pi by 4. So this will be equal to pi into pi by 2 which is equal to pi square upon 2. So i will be equal to pi square upon 4. Thus pi square upon 4 is the required answer to this question. With this we finish this session. Hope you must have enjoyed it. Goodbye, take care and keep smiling.