 Hello friends and how are you all today? The question says integrate the following rational functions. Here the function which is given to us is x upon x minus 1 into x minus 2 into x minus 3. So here you can write the given rational function as a upon x minus 1 plus b upon x minus 2 plus c upon x minus 3. Now on taking LCM we have x upon x minus 1 into x minus 2 into x minus 3 equal to a into x minus 2 into x minus 3 plus b into x minus 1 into x minus 3 plus c into x minus 1 into x minus 2. The whole upon x minus 1 into x minus 2 to x minus 3. Now since we have the same denominator equating the numerators we have x is equal to a on solving we have x square minus 5x plus 6 plus b x square minus 4x plus 3 plus c x square minus 3x plus 2. On rearranging we can write x is equal to if we take x square common we are left with a plus b plus c then if we take x common then we have minus 5a minus 4b minus 3c and taking constants we have 6a 3b and 2c. The respective coefficients we have the equations as plus b plus c is equal to 0 then minus 5a minus 4b minus 3c equal to 1 and the constants equal to 0. Thus on solving these equations we have the value of e as 1 by 2 the value of b is equal to minus 2 and the value of c is obtained to be 3 by 2 that is after simplifying these three equations. So further we can write x upon x minus 1 into x minus 2 into x minus 3 equal to a that is 1 by 2 upon x minus 1 minus 2 upon x minus 2 and then plus 3 by 2 into x minus 3. Now integrating both the sides we have integral of x upon x minus 1 into x minus 2 into x minus 3 into dx equal to 1 upon 2 into x minus 1 minus 2 upon x minus 2 plus 3 upon 2 into x minus 3 into dx. Now taking the integral separately we have equal to 1 by 2 integral of dx upon x minus 1 minus 2 integral of dx upon x minus 2 plus 3 integral of plus 3 by 2 integral of dx upon x minus 3. So we have 1 by 2 log x minus 1 minus 2 log x minus 2 plus 3 by 2 log x minus 3 plus c. So we have the answer as 1 by 2 log mod of x minus 1 minus 2 log mod of x minus 2 plus 3 by 2 log x minus 3 plus c. Right, so this is the required answer understood it well and have a nice day.