 Hi, and welcome to a session where we discuss the following question. The question says, prove that integral of sin x minus cos x by 1 plus sin x cos x from 0 to pi by 2 is equal to 0. Let's now begin with the solution. Let i is equal to integral of sin x minus cos x by 1 plus sin x cos x from 0 to pi by 2. We know that integral of fx from 0 to a is equal to integral of f a minus x from 0 to a. Now by using this property, we get i as integral of sin pi by 2 minus x. Now here a is equal to pi by 2 minus cos pi by 2 minus x by 1 plus sin pi by 2 minus x into cos pi by 2 minus x from 0 to pi by 2. Now this is equal to integral of sin pi by 2 minus x is cos x minus cos pi by 2 minus x is sin x by 1 plus cos x into sin x from 0 to pi by 2. Let's name this equation as equation number 1 and this as 2. On adding 1 and 2, we get plus i equals to 0 pi by 2 integral sin x minus cos x by 1 plus sin x cos x plus integral of cos x minus sin x by 1 plus cos x sin x from 0 to pi by 2. And this is equal to integral of sin x minus cos x plus cos x minus sin x by 1 plus sin x cos x from 0 to pi by 2. This is equal to integral of 0 with respect to x is 0. Now here the lower limit is 0 and upper limit is pi by 2 so this is equal to 0. This implies i is equal to 0. Hence we prove that integral of sin x minus cos x by 1 plus sin x cos x from 0 to pi by 2 is 0. So this completes the session by intake care.