 Hi and welcome to the session. I am Purva and I will help you with the following question. Prove the following integral limit is from 0 to 1 x into e raise to the power x dx is equal to 1. Now we start with the solution. We denote this left hand side by i. So we have i is equal to integral limit is from 0 to 1 x into e raise to the power x dx. Now we take this x as the first term and e raise to the power x as the second term and we apply the Bipart's method. Now I will tell you what is Bipart's method. Suppose we have integral u into v dx where u is the first term and v is the second term. Then by applying Bipart's method we get first that is u into integration of second that is v minus integration derivative of first that is derivative of u into integration of second that is integration v. So this is the Bipart's method. Now we will apply this method in this integral i. So we get i is equal to first that is x into integration of second. Now integration of e raise to the power x is e raise to the power x and limit is from 0 to 1 minus integration derivative of first now derivative of x is 1 into integration of second that is integration of e raise to the power x which is e raise to the power x dx limit is from 0 to 1. This is equal to now putting x as upper limit that is 1 we get 1 into e raise to the power 1 minus now putting x as lower limit we get 0 minus now integration of e raise to the power x is e raise to the power x and the limit is from 0 to 1. This is equal to now 1 into e raise to the power 1 is e raise to the power 1 minus e raise to the power upper limit that is 1 minus e raise to the power lower limit that is 0 and this is for the equal to e raise to the power 1 minus e raise to the power 1 opening the brackets and minus into minus is plus e raise to the power 0. Now cancelling this minus e raise to the power 1 and plus e raise to the power 1 we get this is equal to e raise to the power 0. Now this is equal to e raise to the power 0 is 1. So we have got our answer hence we have proved that integral limit from 0 to 1 x into e raise to the power x dx is equal to 1. Hope you have understood the solution take care and God bless you.