 Hi and welcome to the session. Let us discuss the following question. The question says, by using the properties of definite integrals, calculate the following. Integral of cos2x from 0 to pi by 2. Before solving this question, we should first reverse with second fundamental theorem of integral calculus. In the second fundamental theorem of integral calculus, f pi x is the inter-derivative of continuous function fx defined on closed interval a, b. That is derivative of pi x is equal to fx from a to v is equal to ib minus pi a. You should also know one of the properties of definite integrals which says integral of fx 0 to a is equal to integral of a minus x from 0 to a. Property is the key idea in this question. It is equal to integral of, from the key idea we know that integral of 0 to a is equal to integral of from 0 to a. So using this property integral of from 0 to pi by 2 is equal to integral of x from 0 to pi by 1 is equation number 1 and this from 0 to pi by 2 is equal to find on closed interval a, b. Then integral is equal to pi implies i is equal to