 Hello and welcome to the session. The question is integrate the following function and the given function is root over x square plus 4x minus 5. Let's start with the solution and we have to integrate the function x square plus 4x minus 5 with respect to x. Now if we have a polynomial of the form x square plus bx plus c then it can be written as a into x plus b upon 2a whole square plus c upon a minus b square upon 4a square. So with the help of this let us write x square plus 4x minus 5 as the sum of squares of two polynomials. So we have x plus 2 whole square plus minus 5 upon 1 minus 16 upon 4 into 1 square. This is further equal to 4 into 4 is 16 and we have x plus 2 whole square minus 5 plus 4 is 9 which can be written as 3 square. Thus this integral can further be written as root over x plus 2 whole square minus 3 square into dx. Now let us take t is equal to x plus 2. So this implies dt is equal to dx on differentiating both sides with respect to x. So this integral can be written as root over t square minus 3 square into dt. Now if we have an integral of the type x square minus a square and we have to integrate it with respect to x and its formula is x upon 2 into root over x square minus a square minus a square upon 2 log mod x plus root over x square minus a square plus c. So let us use this integral formula to write integral root over t square minus 3 square into dt. So this we can write it as t upon 2 root over t square minus 3 square minus a square upon 2 a is 3. So we have 9 upon 2 log mod t plus root over t square minus 3 square plus c where c is a constant. Now let us put the value of t which is x plus 2. So we have x plus 2 upon 2 root over t square minus 3 square and x plus 2 whole square minus 3 square and this is for the equal to x square plus 4x minus 5. Since we have derived this from this polynomial only. Now we have minus 9 upon 2 log t is x plus 2 plus on simplifying this we again get root over x square plus 4x minus 5 plus a constant c. Thus on integrating the given function we get x plus 2 upon 2 root over x square plus 4x minus 5 minus 9 upon 2 log mod x plus 2 plus root over x square plus 4x minus 5 plus c. So this completes the session. Hope you have understood it well. Take care and have a nice day.