 Hello and welcome to the session. Let's discuss a question which says evaluate integral from 0 to 2 x square plus 2x plus 1 tx as limit of sums. Now we know that 1 plus 2 plus 3 plus so on n minus 1 is equal to n into n minus 1 over 2 and 1 square plus 2 square plus 3 square plus so on till n minus 1 square is equal to n into n minus 1 into 2 and minus 1 over 6. So this is the key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now by definition of definite integral as the limit of sum we have integral from a to b f of x dx is equal to b minus a into limit n tends to infinity 1 over n into f of a plus f of a plus h plus so on plus f of a plus n minus 1 into h where h is equal to b minus a over n. Now here we have a is equal to 0, b is equal to 2 and f of x is equal to x square plus 2x plus 1. So we can write here a is equal to 0, b is equal to 2 and f of x is equal to x square plus 2x plus 1. So h is equal to 2 minus 0 over n which is equal to 2 over n. Therefore integral from 0 to 2 x square plus 2x plus 1 dx is equal to b minus a which is 2 minus 0 that is 2 into limit n tends to infinity 1 over n into f of a which is f of 0 plus f of a plus h which is f of 2 over n plus f of a plus 2h which is f of 2 into 2 over n plus so on plus f of 2 into n minus 1 over n this is equal to 2 into limit n tends to infinity 1 over n into f of 0 which is 0 plus 0 plus 1 that is 1 plus f of 2 over n which is equal to 2 square over n square plus 2 into 2 over n plus 1 plus f of 4 over n which is equal to 4 square over n square plus 2 into 4 over n plus 1 plus so on plus f of 2n minus 2 over n which is equal to 2n minus 2 whole square over n square plus 2 into 2n minus 2 over n plus 1 this is equal to 2 into limit n tends to infinity 1 over n into 1 plus 1 plus 1 so on plus 1 now these are n terms 1 over n square into 2 square plus 4 square plus so on take 2n minus 2 whole square 2 over n into plus 4 plus so on plus 2n minus 2 and this is equal to 2 into limit n tends to infinity 1 over n into now this sum is equal to n plus now these terms can be written as 2 square over n square into 1 square plus 2 square plus so on plus n minus 1 whole square plus 2 over n into 2 into 1 plus 2 plus so on plus n minus 1 now this is equal to 2 into limit n tends to infinity 1 over n into n plus 4 over n square into now according to our key idea 1 square plus 2 square plus so on plus n minus 1 whole square is equal to n into n minus 1 into 2n minus 1 over 6 plus 4 over n now again according to our key idea 1 plus 2 plus so on plus n minus 1 is equal to n into n minus 1 over 2 now we will solve this expression so this is equal to 2 into limit n tends to infinity 1 over n into n plus 2 over 3n into 2n square minus 3n plus 1 plus 2 into n minus 1 and this is again equal to 2 into limit n tends to infinity 1 plus 2 over 3 into 2 minus 3 over n plus 1 over n square plus 2 into 1 minus 1 over n this is equal to 2 into 1 plus 4 over 3 plus 2 which is again equal to 2 into 13 over 3 and this is equal to 26 over 3 so 26 over 3 is the answer for the above question this completes our session I hope the solution is clear to you bye and have a nice day