 Hello and welcome to the session, the given question says integrate the following function. So first let us learn the formula to integrate a function of the type a square plus x square with respect to x. This is equal to half x into root over x square plus a square plus a square upon 2 log mod x plus root over x square plus a square plus a constant c. So with the help of this formula we shall integrate the given function. So this is our key idea. Now let us start with the solution. So we have to integrate the given function which is x square plus 4x plus 6 with respect to x. Now if we have a polynomial ax square plus bx plus c then can be written as a into x plus b upon 2a whole square plus c upon a minus b square upon 4a square. So let us try to write the polynomial x square plus 4x plus 6 as the sum of the squares of two polynomials. So we have x plus 4 upon 2 whole square plus 6 upon 1 minus b square b square so 4 square is 16 upon 4 into 1 square. This is further equal to x plus 2 whole square plus 4 4s are 16 and 6 minus 4 is 2 so it can be written as root 2 whole square. So the given integral can be written as integral root over x plus 2 whole square plus root over 2 whole square into dx. Now let us take x is equal to x plus 2 so this implies on differentiating both sides with respect to x dt is equal to dx. So the given integral can be written as root over t square plus root 2 whole square into dt. Now by using the key idea this can further be written as half t into root over p square plus 2 plus then we have a square upon 2 in a way is root 2 so root 2 whole square is 2 upon 2 log mod t plus root over t square plus root 2 whole square which is 2 plus a constant c. Now let us put the value of t so we have half into x plus 2 into root over x plus 2 whole square plus 2 is equal to x square plus 4x plus 6 plus 2 cancels out with 2 and we have log mod t is x plus 2 and again root over t square plus 2 is equal to root over x square plus 4x plus 6 plus a constant c. Thus on integrating the given function we get x plus 2 upon 2 into root over x square plus 4x plus 6 plus log mod x plus 2 plus root over x square plus 4x plus 6 plus c. So this completes the session. Bye and take care.