 Hello and welcome to the session. Let us discuss the following question. Question says evaluate the following integral. Given integral is definite integral from 0 to pi upon 2 sin x upon 1 plus cos square x dx. First of all let us understand that integral 1 upon 1 plus x square dx is equal to tan inverse x plus c. This formula of integration is the key idea to solve the given question. Let us now start with the solution. Now we have to find the integral from 0 to pi upon 2 sin x upon 1 plus cos square x dx. Now we know derivative of cos x is minus sin x. So we will solve this integral by substitution method. So let us put cos x is equal to t. Now differentiating both the sides with respect to x we get minus sin x dx is equal to dt. Now we get sin x dx is equal to minus dt. Now the new limits are when x is equal to 0 t is equal to 1 and when x is equal to pi upon 2 then t is equal to 0. Now given definite integral that is integral from 0 to pi upon 2 sin x upon 1 plus cos square x dx is equal to definite integral from 1 to 0 minus dt upon 1 plus t square. Now this integral is equal to definite integral from 0 to 1 dt upon 1 plus t square. Here we have used this property of definite integral. We know definite integral from a to b fx dx is equal to definite integral from b to a fx dx multiplied by minus 1. Now using the formula of integration given in t idea we get this integral is equal to tan inverse t where limits of t are from 0 to 1. Now substituting the limits in this function we get tan inverse 1 minus tan inverse 0. We know tan inverse 1 is equal to pi upon 4 and tan inverse 0 is equal to 0. So we get pi upon 4 minus 0. Now this is further equal to pi upon 4. So we get definite integral from 0 to pi upon 2 sin x upon 1 plus cos square x dx is equal to pi upon 4. So this is our required answer. This completes the session. Hope you understood the solution. Take care and keep smiling.