 Hello and welcome to the session. The given question says, evaluate integral dx divided by root over x square minus 3x plus 2. So let's start with the solution. And let us denote the given integral by i. So we have i is equal to integral dx divided by root over x square minus 3x plus 2. Now if we have a polynomial of the type A x square plus dx plus c, then it can be written as the square of two polynomials. And that's why A into x plus b divided by 2a whole square plus c divided by a minus b square divided by 4a square and this can be written as root over into whole square. So that's why we have written it in the form of square of two polynomials. So let us write x square minus 3x plus 2 as the square of two polynomials. So we have x then we have plus b divided by 2a. So here we have minus 3 divided by 2 whole square plus then we have c divided by a that is 2 divided by 1 minus b square this is 9 divided by 4. So this is further equal to x minus 3 divided by 2 whole square plus on simplifying this we get minus 1 divided by 4 which can be written as x minus 3 divided by 2 whole square minus 1 divided by 2 whole square. So this integral can further be written as integral dx divided by root over x minus 3 by 2 whole square minus half square. Now let us say t is equal to x minus 3 by 2. So this implies dt is equal to dx thus integral I have written as integral dx divided by root over t square minus half square. Now this is in the form of integral dt divided by t square minus a square and this is equal to log mod t plus root over t square minus a square plus a constant c. So by using this I can further written as log mod t plus root over t square minus a square plus a constant c. This is further equal to log now t we have assumed as x minus 3 by 2 plus t square minus a square is x minus 3 by 2 whole square minus half square and this in turn is equal to x minus 3x plus 2. So here we have x minus 3x plus 2 plus a constant c and this is further equal to log mod minus 3 divided by 2 plus x square minus 3x plus 2 plus a constant c. Thus on evaluating we get an answer log mod 2x minus 3 divided by 2 plus x square minus 3x plus 2 plus a constant c. So this completes the