 Hello and how are you all today? My name is Priyanka and I shall be helping you with the following question. See, integrate the following rational functions. Now, the function which we need to integrate is x upon x minus 1 the whole square multiplied by x plus 2. Now here, we can see that the integrant is a proper rational function. So decomposing the rational function into partial fractions we get x upon x minus 1 the whole square into x plus 2 is equal to a upon x minus 1 plus p upon x minus 1 the whole square plus c upon x plus 2. Now on taking the LCM, we have x upon x minus 1 the whole square into x plus 2 in the left hand side is equal to a into x minus 1 the whole square. Sorry, x minus 1 into x plus 2 plus b into x plus 2 and plus c into x minus 1 the whole square. Hold upon x minus 1 into x minus 2 the whole square into x in the whole divided by x minus 1 the whole square into x plus 2 that is the LCM. Now, putting the value of x equal to 1 in the above equation numerator we get 1 is equal to a 1 minus 1 multiplied by 1 plus 2 plus p into 1 plus 2 plus c 1 minus 1 the whole square that is giving us the value of p as 1 by 3. Now, putting the value of x is equal to minus 2 we get in this equation itself that is the numerator we have minus 2 is equal to a minus 2 minus 1 into minus 2 plus 2 this term will become 0 plus b minus 2 plus 2 this term will again become equal to 0 plus c minus 2 minus 1 the whole square. So, we have the value of c as minus 2 by 9 and lastly on comparing the coefficients of x square in both sides we get 0 is equal to a plus c that is the value of a is minus c that implies the value of a is minus of minus 2 by 9 that gives us the value as 2 by 9. So, we can write the function x upon x minus 1 the whole square into x plus 2 is equal to a that is 2 by 9 into x plus x minus 1 plus 1 upon 3 into x minus 1 the whole square minus 2 by 9 into x plus 2. Now, integrating both the sides we have x upon x minus 1 the whole square into x plus x plus 2 the whole square into dx that is not in whole square is equal to integral of 2 upon 9 x minus 1 into dx plus dx upon 3 into x minus 1 the whole square minus 2 upon 9 x plus 2 dx. So, we have the answer as 2 by 9 log of mod x minus 1 plus 1 upon 3 it will be x minus 1 raised to the power minus 1 upon minus 1 minus 2 upon 9 log x plus 2 plus c. So, we can write the answer as 2 by 9 log x minus 1 upon x plus 2 minus 1 upon 3 into x minus 1 plus c and this is the answer to the question. Hope you enjoyed and understood it too. Have a nice day.