 Hello and welcome to the session, the question says, integrate the following function and the fourth one is root over x square plus 4x plus 1. So let's start with the solution and we have to integrate the given function x square plus 4x plus 1 with respect to x. So first let us learn that if we have a polynomial of the type A x square plus px 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, right? So let us try to write x square plus 4x plus 1 as the sum of squares of two polynomials. So we have x plus 4 upon 2 whole square plus c upon A that is 1 minus b square upon 4a square. So we have 16 upon 4. So this is for the equal to x plus 2 whole square and solving this we have 4 and 1 minus 4 is minus 3. So plus into minus is minus and 3 can be written as root 3 whole square. Thus this integral can further be written as root over x plus 2 whole square minus root 3 whole square into dx. Now let us put x is equal to x plus 2. So this implies dt is equal to dx. So this integral can further be written as root over t square minus root over 3 whole 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. Now let us formalize x upon 2 into root over x square minus a square upon 2 log mod x plus root over x square minus a square plus c. So we shall be using this formula of integral to write this. So this is equal to t upon 2 into root over t square minus 3 minus a square upon 3 that is 3 upon 2 since we have a is equal to root 3. So root 3 square is 3 log mod t plus root over t square minus 3 plus a constant c. This is further equal to p is x plus 2 upon 2 root over t square minus 3 is equal to x plus 2 whole square minus root 3 whole square and this is equal to x square plus 4 x plus 1 minus 3 upon 2 log mod x plus 2 plus root over x square plus 4 x plus 1 plus a constant c. Thus on integrating the given function we get x plus 2 upon 2 root over x square plus 4 x plus 1 minus 3 upon 2 log mod x plus 2 plus root over x square plus 4 x plus 1 plus c. So this completes this equation. Bye and take care.