 Hello and welcome to the session, I am Deepika here. Let's discuss the question which says, integrate the column function x tan inverse x. Now we know the integration of product of two functions that is integral of fx dx is equal to first function that is fx integral of the second function that is gx minus integral of differential coefficient of the first function that is f dash x into integral of the second function. So this is a key idea behind our question. We will take the help of this key idea to solve the above questions. So let's start the solution. Now we have to integrate x tan inverse x. Now whenever we have to integrate product of two functions and one function is inverse trigonometric function then we will take the inverse trigonometric function as the first function. So here x is our second function and tan inverse x is our first function. So integral of x tan inverse x is equal to first function which is tan inverse x into integral of the second function that is x minus integral of derivative of the first function that is dy dx of tan inverse x integral of and this is equal to tan which is 1 over 1 plus x square integral of x which is x square upon 2 is equal to 2 upon 2 plus 1 by 2 over 1 plus x square in equal to 2 plus 1 by 2. Now we will integrate x square upon 1 plus x square. Now the integral is not a proper regular function. So we will divide x square by 1 plus x square. So we will have 1 minus 1 over 1 plus x square. This is again equal to x square by 2 tan inverse 1 by 2 integral of 1 plus 1 by 2 integral of 1 over 1 plus x square and this is equal to x square by 2 tan inverse integration and integration are inverses of each other. Now derivative of tan inverse x is 1 over 1 plus x square or 1 over 1 plus x square is tan inverse x. So this is plus 1 by 2 tan inverse x plus 1 by 2 tan x solution is clear to you. Bye and take care.