 Hi, and welcome to our session. Let us discuss the following question. The question says, by using the properties of definite integrals, evaluate the following. Integral of cos x to the power 5 from 0 to 2 pi. Before solving this question, we should know that integral of fx from 0 to 2a is equal to 2 times integral of fx from 0 to a, if f of 2a minus x is equal to fx. And this integral is equal to 0 if of 2a minus x is equal to minus fx. You should also know that integral of fx from 0 to a is equal to integral of fa minus x from 0 to a. The knowledge of these two properties are the key ideas in this question. Let's now begin with the solution. Now, in this question, we have to evaluate integral of cos x to the power 5 from 0 to 2 pi. Now, here the upper limit is of the form 2a. So we will first apply the first property. Now, let fx is equal to cos x to the power 5. Now, f2 pi minus x is equal to cos 2 pi minus x whole to the power 5. Now, cos 2 pi minus x is cos x. So this is equal to cos x to the power 5. And this is equal to cos x to the power 5. And this is equal to fx. 2 pi minus x is equal to fx. Therefore, fx is an even function. From the key idea, we know that if f of 2a minus x is equal to f of x, then integral of fx from 0 to 2a is equal to integral of fx from 0 to a into 2. So by using property integral of cos x to the power 5 from 0 to 2 pi is equal to 2 times into integral of cos x to the power 5 from 0 to pi. 2a is equal to 2 pi. Therefore, a is equal to pi. Now, again, let gx is equal to cos x to the power 5. So let's use the second property, as here the limits are from 0 to pi. Now, g of pi minus x is equal to cos pi minus x whole to the power 5. Cos pi minus x is minus cos x. So we have minus cos x to the power 5. And this is equal to minus cos x to the power 5. And this is equal to minus gx. As d of pi minus x is equal to minus gx, so this implies gx is an odd function. 2 times integral of cos x to the power 5 from 0 to pi is equal to 2 into 0. And this is equal to 0. Hence, our required answer is 0. So this completes the session. Bye and take care.