 Hello friends and how are you all? My name is Priyanka and let us discuss this question, says integrate the following rational function. Now here the function which is given to us is xq plus x plus 1 upon x square minus 1. Now here the integrand is not a proper rational function. So let us first simplify the given fraction by long division. So we have to simplify x cube plus x plus 1 upon x square minus 1 with the help of long division method. Now here when multiplying it by x we have x cube minus x. So on solving we are left with 2x plus 1, right? So we can write this given function as x cube plus x plus 1 upon x square minus 1 can be written as x plus 2x plus 1 upon x square minus 1, right? Further, now we need to simplify this function separately. Now the given function is 2x plus 1 upon x square minus 1 it can be written as 2x plus 1 upon x plus 1 into x minus 1. Now further 2x plus 1 upon x plus 1 to x minus 1 using partial fractions can be written as a upon x plus 1 plus b upon x minus 1. Taking the LCM we have a is equal to x minus 1 plus b into x plus 1 and equating the numerator with the numerator itself. We have the value of a coming out to be 1 by 2 that is after simplification and the value of b is coming out to be 3 by 2. So here this function can be written as 1 upon 2 into x plus 1 plus 3 upon 2 to x minus 1. So let us write it in place of the given function. We have the given function as x plus 1 upon 2 x plus 1 plus 3 upon 2 x minus 1 that is x cube plus x plus 1 upon x square minus 1. Now integrating both sides we have integral of x cube plus x plus 1 upon x square minus 1 dx is equal to integral of dx. x dx plus 1 by 2 integral of dx upon x plus 1 plus 3 by 2 integral of dx upon x minus 1. Now this can be further written as. Now here the integral of x dx is what exactly x square by 2 plus 1 by 2 log mod of x plus 1 plus 3 by 2 log mod of x minus 1 plus c. So the required answer to this question is x square by 2 plus 1 by 2 log mod x plus 1 plus 3 by 2 log mod x minus 1 plus c. So this ends the question. Hope you understood the whole concept well. First of all factorize the given function and then with the help of the given two functions that we obtained after long division method, split them into partial fractions and then integrate by using the methods that is known to you, right? So have a good day ahead.