 Hi and welcome to our session. Let us discuss the following question. The question says, are using the properties of definite integrals evaluate the pollen? Integral of sine square x from minus pi by 2 to pi by 2. Before solving this question, we should note two properties of definite integrals. First property of definite integrals is integral of fx from 0 to a is equal to integral of f a minus x from 0 to a. And second property of definite integrals is integral of fx from minus a to a is equal to 2 times integral of fx from 0 to a. If f is an even function, it is equal to f of minus x. Integral of fx from minus a to a is equal to 0. If f is an odd function, f of minus x is equal to minus fx. The knowledge of these two properties are the key ideas in this question. Let x is equal to sine square x. We will first take that whether sine square x is an even function or an odd function. If we replace x by minus x, then we get sine minus x whole square. We know that sine minus x is minus sine x. So we have minus sine x whole square and this is equal to sine square x. And sine square x is fx. So f of minus x is equal to fx and this implies fx is an even function. i equals to integral of sine square x from minus pi by 2 to pi by 2 is equal to 2 times integral of sine square x from 0 to pi by 2 into 2. From the key idea, we know that integral of fx from 0 to a is equal to integral of f a minus x from 0 to a. By using this property, we get i as 2 into integral of sine square pi by 2 minus x from 0 to pi by 2. This is equal to 2 times integral of cos square x from 0 to pi by 2. Let's say this equation is equation number 1. And this as 2. And adding 1 and 2, we get equals to 2 into integral of sine square x from 0 to pi by 2 plus 2 times integral of cos square x from 0 to pi by 2. This is equal to 2 times integral of cos square x plus sine square x from 0 to pi by 2. We know that cos square x plus sine square x is equal to 1. So we have 2 into integral of 1 from 0 to pi by 2. Now integral of 1 with respect to x is x. So we have 2x lower limit is 0 and upper limit is pi by 2. Now by using second fundamental theorem of integral calculus, we have 2 into pi by 2 minus 2 into 0. We have first substituted upper limit in place of x and then we have substituted lower limit in place of x. Now this is equal to pi. So 2i is equal to pi and this implies i is equal to pi by 2. Hence our required answer is pi by 2. So this completes the association i and take care.