 Hello and welcome to the session the question says, integrate the following function and the first minus root over 1 minus 4x minus x square. So, let us start with the solution we have to integrate the given function 1 minus 4x minus x square root over with respect to x. First let us learn that if we have a polynomial x square plus bx plus c then we can write it as a into x plus b upon 2a whole square plus c upon a minus b square upon 4a square. So, 1 minus 4x minus x square can be written as minus of x square plus 4x minus 1 or minus x plus 2 whole square plus minus 1 upon 1 minus 16 upon 4. We can further be written as now simplifying this bracket here we have minus 5 and by opening this square bracket we get plus 5 which can be written as root 5 whole square minus x plus 2 whole square. So, this integral can further be written as integral root over root 5 whole square minus x plus 2 whole square into dx. Now let us put t is equal to x plus 2. So, this implies dt is equal to dx. Thus this integral can further be written as root over root over 5 whole square minus t square into dt. Now if we have a integral of the type root over a square minus x square and we have to integrate it with respect to x then its formula is half into x root over a square minus x square plus a square upon 2 sin inverse x upon a plus c. So, this integral can further be written as half into t root over root over 5 whole square minus t square plus a square upon 2. So, a square is root over 5 whole square upon 2 sin inverse x upon a. So, x is t upon root over 5 plus a constant c. This is further equal to now let us put the value of t and t is x plus 2. So, we have x plus 2 upon root over 5 whole square minus t square and this is equal to root over 5 whole square minus x plus 2 whole square since t is equal to 2 and this is further equal to 1 minus 4 x minus x square. So, here we have 1 minus 4 x minus x square plus 5 upon 2 sin inverse x plus 2 upon root over 5 plus c. Thus on integrating the given function we get x plus 2 upon 2 root over 1 minus 4 x minus x square plus 5 upon 2 sin inverse x plus 2 upon root over 5 plus c or it can also be written as 5 upon 2 sin inverse x plus 2 upon root over 5 plus x plus 2 upon 2 root over 1 minus 4 x minus x square plus c. So, this completes the session by interview.