 Hello and welcome to the session. Let us discuss the following question. Question says, by using the properties of definite integrals, evaluate the given integral. Given integral is definite integral from 0 to pi upon 2 root sin x upon root sin x plus root cos x dx. First of all, let us understand that definite integral from 0 to a fx dx is equal to definite integral from 0 to a f a minus x dx. This is the key idea to solve the given question. Let us now start with the solution. We are given definite integral 0 to pi upon 2 root sin x upon root sin x plus root cos x dx. Now this integral can be written as definite integral from 0 to pi upon 2 square root of sin pi upon 2 minus x upon square root of sin pi upon 2 minus x plus square root of cos pi upon 2 minus x dx. Here we have used the property given in key idea that definite integral from 0 to a fx dx is equal to definite integral from 0 to a f a minus x dx. Here value of a is pi upon 2. So for every x we can write pi upon 2 minus x. Now we know sin pi upon 2 minus x is equal to cos x. So we can write this definite integral as definite integral from 0 to pi upon 2 square root of cos x upon square root of cos x plus now here cos pi upon 2 minus x is equal to sin x. So here we can write sin x. Now let us assume that given integral is equal to i and here we get i is equal to this integral. Now let us name this expression as 1 and this expression as 2. Now adding expression 1 and 2 we get 2i is equal to integral from 0 to pi upon 2 root sin x plus root cos x upon root cos x plus root sin x dx. Now numerator and denominator of this integrand will get cancelled with each other and we get 2i is equal to definite integral from 0 to pi upon 2 dx. Now we know integral of 1 with respect to x is equal to x only and our given limits are 0 to pi upon 2. Now substituting the limits in the answer we get 2i is equal to pi upon 2 minus 0. This further implies 2i is equal to pi upon 2 or we can say i is equal to pi upon 4. Now we know i is equal to integral 0 to pi upon 2 root sin x upon root sin x plus cos x dx. So we get this integral is equal to pi upon 4. This is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.