 Hello and welcome to the session. The question says, integrate the following function and the last question is 5x plus 3 upon root over x square plus 4x plus 10. So let's start with the solution. Therefore we have 5x plus 3 upon root over x square plus 4x plus 10 and we have to integrate this function with respect to x. Let us say 5x plus 3 is equal to a into the derivative of the polynomial x square plus 4x plus 10 plus 3. Therefore 5x plus 3 is equal to a into derivative of x square plus 4x plus 10 is 2x plus 4 plus 3. So that's in place on equating the coefficients of the variable x and the constants. We have 5 is equal to 2a and 4a plus b is equal to 3. So from here it implies that a is equal to 5 upon 2 and from here on simplifying we get b is equal to minus 7. Now so this integral can further written as integral 5x plus 3 is equal to a into 2x plus 4 plus b and the denominator we have root over x square plus 4x plus 10 into dx. Now putting the values of a and b we have 5 upon 2 into 2x plus 4 plus b is minus 7 upon root over x square plus 4x plus 10 and we have to integrate it with respect to x. So now this can further written as taking the constants outside the integral saying we have 5 upon 2 integral 2x plus 4 into dx upon root over x square plus 4x plus 10 minus 7 times of dx upon root over x square plus 4x plus 10. Now let us solve both these integrals one by one. First it is solve integral 2x plus 4 into dx upon root over x square plus 4x plus 10. Let us take t is equal to x square plus 4x plus 10. So this implies dt is equal to 2x plus 4 into dx. So this integral can further written as dt upon root over t. This is equal to 2 times root over t plus a constant c1 since to integrate a function x raised to the power n with respect to x it is equal to x raised to the power n plus 1 upon n plus 1 plus a constant c1. So this can further written as 2 root value of t is x square plus 4x plus 10 plus a constant c1. So this can further written as 5 upon 2 into 2 root over x square plus 4x plus 10 plus a constant c1 then we have minus 7 and now let us find the value of this integral. Now this is integral dx upon root over x square plus 4x plus 10. Now polynomial of the type ax square plus bx plus c can be written as a into x plus b upon 2a whole square plus c upon a minus b square upon 4a square. Therefore polynomial x square plus 4x plus 10 can be written as x plus 4 upon 2 whole square plus c upon n minus b square upon 4a square. So b is 4 so 4 square is 16 upon 4 into a square that is 1 square. So this is further equal to x plus 2 whole square. Now 4 force is 16 and 10 minus 4 is 6. So we have plus root 6 whole square. So this can be written as integral dx upon root over x plus 2 whole square plus root 6 whole square. Now let us take t is equal to x less 2. So this implies dt is equal to dx. So this integral can further be written as dt upon root over t square plus root over 6 whole square. Now if we have an integral of the type 1 upon root over x square plus a square then its value is equal to log mod x plus root over x square plus a square plus a constant. So this can be written as log mod t plus root over t square plus root 6 whole square is 6 square plus a constant let c2. So it can further be written as log t is x plus 2 plus root over t square plus 6 square is x plus 2 whole square plus root 6 whole square sorry here also we have root 6 square therefore here we can write it as x square plus 4x plus 10 since x square plus 4x plus 10 is equal to x plus 2 whole square plus root 6 whole square plus a constant c2. So here we can write that log mod x plus 2 plus root over x square plus 4x plus 10 plus a constant c2. This is further equal to 2 cancels output 2 here 5 root over x square plus 4x plus 10 minus 7 log mod x plus 2 plus root over x square plus 4x plus 10 plus c1 minus 7 c2 and this is a constant let us denote it by c thus on integrating the given function we get 5 root over x square plus 4x plus 10 minus 7 log mod x plus 2 plus root over x square plus 4x plus 10 plus a constant c. So this completes the session by intake care.