 Hi and welcome to the session. Let us discuss the following question. The question says prove that integral of fx from a to v is equal to integral of f a plus v minus x from a to v. Hence, did you start integral of 1 by 1 plus root tan x from pi by 6 to pi by 3 is equal to pi by 12. Solution, we will first prove the first part. Let us put x equals to a plus v minus t. Now differentiating this with respect to t, we get dx by dt equals to minus 1. This implies dx is equal to minus dt. Now, when x is equal to a, then t is equal to v and when x is equal to v is equal to a. So, integral of fx from a to v is equal to minus into integral of a plus v minus t from v to a and this is equal to integral of f a plus v minus x from a to v. Hence, we prove that integral of fx from a to v is equal to integral of f a plus v minus x from a to v. Now, we will prove the second part. Let i is equal to integral of 1 by 1 plus root tan x from pi by 6 to pi by 3. Now, this is equal to integral of root cos x divided by root cos x plus root sin x from pi by 6 to pi by 3. Now, by using the above result that is integral of fx from a to v is equal to integral of f a plus e minus x from a to v. We get i as integral of root cos pi by 3 plus pi by 6 minus x divided by root cos pi by 3 plus pi by 6 minus x plus root sin pi by 3 plus pi by 6 minus x from pi by 6 to pi by 3. Now, this is equal to integral of cos pi by 2 minus x divided by root sin pi by 2 minus x plus root cos pi by 2 minus x from pi by 6 to pi by 3. Now, this is equal to integral of root sin x divided by root cos x plus root sin x from pi by 6 to pi by 3. i is equal to integral of root sin x divided by root cos x plus root sin x from pi by 6 to pi by 3. Let us name this equation as equation number 1 and this as equation number 2. Now, we will add equation 1 and 2. So, we get i equals to integral of root cos x divided by root sin x plus root cos x from pi by 6 to pi by 3 plus integral of root sin x divided by root cos x plus root sin x from pi by 6 to pi by 3. This is equal to integral of root cos x plus root sin x divided by root cos x plus root sin x from pi by 6 to pi by 3. And this is equal to integral dx from pi by 6 to pi by 3. So, 2i is equal to integral dx from pi by 6 to pi by 3. Now, this implies 2i is equal to x, lower limit is pi by 6 and upper limit is pi by 3. This implies 2i is equal to pi by 3 minus pi by 6. This implies i is equal to pi by 6 and this implies i is equal to pi by 12. x integral of 1 by 1 plus root tan x from pi by 6 to pi by 3 is equal to pi by 12. So, this completes the solution i and take care.