 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 sine x to the power 3 by 2 by sine x to the power 3 by 2 plus cos x to the power 3 by 2 from 0 to 5 by 2. Following this question, we should first be well versed with second fundamental theorem of integral calculus. According to second fundamental theorem of integral calculus, if phi x is the anti-derivative of continuous function fx defined on close interval a, b, that is derivative of phi x is equal to fx, then integral of fx from a to b is equal to phi b minus phi a. Also know one of the properties of definite integrals which says integral of fx from 0 to a is equal to integral of f of a minus x from 0 to a. For the help of this theorem and this property, we will solve this question. So always remember this. Now begin with the solution. Let i is equal to integral of sine x to the power 3 by 2 by sine x to the power 3 by 2 plus cos x to the power 3 by 2 from 0 to pi by 2. We know that integral of fx from 0 to a is equal to integral of f of a minus x from 0 to a. By using this property, this integral is equal to, now here a is equal to pi by 2, so this is equal to integral of sine pi by 2 minus x to the power 3 by 2 by sine pi by 2 minus x to the power 3 by 2 plus cos pi by 2 minus x to the power 3 by 2 from 0 to pi by 2. We know that sine pi by 2 minus x is cos x, so this is equal to integral of cos x to the power 3 by 2 by cos x to the power 3 by 2 plus sine x to the power 3 by 2 from 0 to pi by 2. So now i is equal to integral of cos x to the power 3 by 2 by cos x to the power 3 by 2 plus sine x to the power 3 by 2 from 0 to pi by 2. Let's name this equation as equation number 1 and this as 2. Now on adding 1 and 2, we get i is equal to integral of sine x to the power 3 by 2 by sine x to the power 3 by 2 plus cos x to the power 3 by 2 from 0 to pi by 2 plus cos x to the power 3 by 2 by cos x to the power 3 by 2 plus sine x to the power 3 by 2 from 0 to pi by 2. Now this is equal to integral of sine x to the power 3 by 2 plus cos x to the power 3 by 2 by sine x to the power 3 by 2 plus cos x to the power 3 by 2 from 0 to pi by 2. Now this implies 2i is equal to integral of 1 from 0 to pi by 2. This implies i is equal to 1 by 2 into integral of 1 from 0 to pi by 2. We know that integral of 1 with respect to x is x. So this is equal to 1 by 2x, where the lower limit is 0 and upper limit is pi by 2. By the second fundamental theorem, we know that if pi x is the antiderivative of fx, then integral of fx from a to b is equal to pi of b minus pi of a. Now here x is the antiderivative of 1. That means here pi is x and f is 1. So by the second fundamental theorem, this is equal to 1 by 2 into pi by 2 minus 1 by 2 into 0. This is equal to pi by 4. Hence our required answer is pi by 4. So this completes the session. Bye and take care.