 Hi and welcome to the session. Let us discuss the following question. Question says, by using properties of definite integrals, evaluate the following integral. Given integral is, from 0 to pi upon 2 sin x minus cos x upon 1 plus sin x multiplied by cos x dx. First of all, let us understand that integral from 0 to a fx dx is equal to 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. Now we have to find the definite integral from 0 to pi upon 2 sin x minus cos x upon 1 plus sin x multiplied by cos x dx. Let us assume that this definite integral is equal to i. Now using the property given in key idea, we can write this integral as 0 to pi upon 2 sin pi upon 2 minus x minus cos pi upon 2 minus x upon 1 plus sin pi upon 2 minus x multiplied by cos pi upon 2 minus x dx. Now we know sin pi upon 2 minus x is equal to cos x and cos pi upon 2 minus x is equal to sin x. So we can write this integral as cos x minus sin x upon 1 plus cos x multiplied by sin x dx. Now let us name this expression as 1 and this expression as 2. Now adding expressions 1 and 2 we get 2y is equal to definite integral from 0 to pi upon 2 sin x minus cos x plus cos x minus sin x upon 1 plus sin x multiplied by cos x. dx now minus cos x and plus cos x will cancel each other and sin x minus sin x will also cancel each other and we get 2y is equal to 0. Clearly we can see integrand is equal to 0 so this integral is also equal to 0 or we can say i is equal to 0. Now we know i is equal to given definite integral so this integral is equal to 0. So this is our required answer this completes the session hope you understood the solution take care and have a nice day.