 Hi and welcome to the session. I am Purva and I will help you with the following question. Evaluate the definite integral dx upon under root 1 plus x minus under root x where the limit is from 0 to 1. Now we start with this solution. So we denote this integral by i. So we have i is equal to integral 0 to 1 dx upon under root 1 plus x minus under root x. This is equal to integral limit from 0 to 1 1 by under root 1 plus x minus under root x. So rationalizing this we get into under root 1 plus x plus under root x upon under root 1 plus x plus under root x dx. This is equal to integral limit from 0 to 1 under root 1 plus x plus under root x upon. Now we know that a minus b into a plus b is equal to a square minus b square. So we can write this as 1 plus x minus x dx since a minus b into a plus b is equal to a square minus b square. We cancel this plus x and minus x. Now we get this is equal to integral limit 0 to 1 under root 1 plus x plus under root x dx. This is equal to, now we can write this as integral limit from 0 to 1 under root 1 plus x dx plus integral limit 0 to 1 under root x dx. This is further equal to, now integrating under root 1 plus x we get 1 plus x to the whole power 1 by 2 plus 1 upon 1 by 2 plus 1 and limit is from 0 to 1 plus, now integrating under root x we get x to the power 1 by 2 plus 1 by 1 by 2 plus 1 again the limit is from 0 to 1. This is equal to 1 plus x to the power. Now 1 by 2 plus 1 is 3 by 2 upon 3 by 2 the limit is from 0 to 1 plus x to the power 1 by 2 plus 1 is x to the power 3 by 2 upon 3 by 2 again the limit is from 0 to 1. This is equal to now 1 upon 3 by 2 can be written as 2 by 3 into 1 plus x to the power 3 by 2 limit is from 0 to 1 plus again 1 upon 3 by 2 can be written as 2 by 3 into x raise to the power 3 by 2 limit is from 0 to 1. This is equal to 2 by 3 into 1 plus now upper limit is 1 so we will put 1 in place of x we get 1 to the power 3 by 2 minus 2 by 3 into 1 plus now the lower limit is 0 so we will put 0 in place of x we get 0 to the whole power 3 by 2 plus 2 by 3 into now again the upper limit here is 1 so we will put 1 in place of x and 1 raise to the power 3 by 2 is 1 so we will get only 2 by 3 minus 2 by 3 now the lower limit is 0 so putting 0 in place of x this will give us 0 now this is equal to 2 by 3 into 1 plus 1 is 2 so 2 raise to the power 3 by 2 minus 2 by 3 now 1 plus 0 is 1 and 1 raise to the power 3 by 2 is 1 so plus 2 by 3 and 2 by 3 into 0 is 0 so cancelling this 2 by 3 and this 2 by 3 we get this is equal to 2 by 3 into 2 raise to the power 3 by 2 we can also write this as this is equal to 4 by 3 into root 2 so we will write our answer as 4 by 3 into root 2 so this is our answer hope you have understood the solution take care and God bless you