 Hello and welcome to the session. My name is Bansi and I am going to help you with the following question. The question says integrate the following function x into log x the whole square. So let us see the solution to this one. Let I be equal to integral x into log x the whole square dx. This is equal to integral log x the whole square into x dx. Now we see that according to the i-lay true logarithmic function is given preference over algebraic function to be chosen as the first function. So this will be the first function, this will be the second function. Now applying integration by paths we have first function that is log x the whole square into integral of second function that is x. So we have x dx minus integral of dy dx of first function that is log x the whole square into integral x dx the whole into dx. This is equal to log x the whole square into x square by 2 minus integral dy dx of log x the whole square will be 2 into log x into 1 by x into x square by 2 that is integral of x dx into dx. Now this happens because first of all we see that dy dx of x raise to power n is equal to n into x raise to power n minus 1. So first of all we will have 2 into log x raise to power 2 minus 1 is 1 into differentiation of log x that is 1 by x. Now we see that 2 gets cancelled with 2 one of the x gets cancelled one of the x in the numerator we will have x square by 2 log x the whole square minus log x into x dx this is minus integral of log x into x dx again applying by paths this will be the first function this will be the second function we get x square by 2 log x the whole square minus first function that is log x into integration of second function that is x square by 2 minus integration of derivative of first function is 1 by x into integration of x dx is x square by 2 dx this will be equal to x square by 2 into log x the whole square minus x square by 2 log x minus this x gets cancelled with this x and we will have 1 by 2 integration of x dx will be x square by 2 plus some constancy this will be equal to x square by 2 log x the whole square minus x square by 2 log x plus 1 by 2 into x square by 2 plus c this is equal to x square by 2 into log x the whole square minus x square by 2 log x plus 1 by 4 x square plus c so our answer to this question is x square by 2 log x the whole square minus x square by 2 log x plus x square by 4 plus c so i hope that you understood the question and enjoyed the session have a good day