 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says integrate the following function x square plus 1 into log x. So let us start with a solution to this question. Let i be equal to integral of x square plus 1 log x dx. We have to find i in this case. Now we see that we integrate this by parts. Here we see that this is an algebraic function and this is a logarithmic function. According to the i-late rule logarithmic function is given preference over algebraic function for becoming the first function. So log x will be the first function and x square plus 1 will be the second function. Now by integration by parts we have i will be equal to first function that is log x into integral of second function that is x square plus 1 dx minus integral of differentiation of first function that is d by dx of log x into integral of second function that is x square plus 1 dx the whole into dx. This will be equal to log x into now integral of x square plus 1 will be same as integral of x square with respect to x plus integral of 1 with respect to x. So that will be x cube by 3 plus x minus integral of d by dx of log x is 1 by x. Again integral of x square plus 1 dx we have seen here was x cube by 3 plus x into dx. Now this we get because we know that integral x raise to power n dx is equal to x raise to power n plus 1 divided by n plus 1 where n is not equal to minus 1 we add a constant here also. So in this case n was equal to 2 so we get x raise to power n plus 1 that is 2 plus 1 equal to 3 divided by 3 in this case n was 0 because 1 can be written as x raise to power 0. So we get this now this can be further written as x cube by 3 plus x into log x minus now we see that this can be written as integral of x square by 3 plus 1 into dx because 1 by x into x cube by 3 gives us x square by 3 and 1 by x into x gives us 1. This will be equal to x cube by 3 plus x into log x minus x cube by 9 minus x plus c. This again we get by this formula. So we see that this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.