 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says integrate the following function as x cos inverse x divided by square root of 1 minus x square. So let us start with the solution to this question. Let I be equal to integral of cos x cos inverse x divided by square root of 1 minus x square dx. Now what we do here is we put cos inverse x to be equal to t. So we will have minus 1 by square root of 1 minus x square dx will be equal to dt. What we have done is we found derivatives on both the sides. Now derivative of cos inverse x is minus 1 by square root of 1 minus x square and derivative of t with respect to x will be dt by dx. So we have taken dx on the left hand side so we get this. Now putting these values here we get i to be equal to minus integral t cos t dt because 1 upon square root of 1 minus x square dx becomes minus dt minus sign comes out dt we have here. Cos inverse x was equal to t so we have t. Since cos inverse x was equal to t so this implies x is equal to cos t. So we have it like this. Now we apply integration by parts. This will be the first function this will be the second function. This will be equal to minus of first function that is t into integral of cos t dt that is second function minus integral of d by dt of first function that is t into integral of cos t dt into dt. Now this will be equal to minus of t into sin t minus integral of dt by dt will be 1 into sin t dt plus some constancy this will be equal to minus t sin t. Now here we will get minus cos t plus c because integral of sin t dt is minus cos t minus cos t with minus becomes plus and we have a negative sign here so we have a negative sign here. Now this is equal to minus t now sin t is square root of 1 minus cos square t minus cos t plus c. Now we have this in the terms of t but we want the arms in terms of x so we will convert it to x and we get t was equal to cos inverse x so we have minus cos inverse x square root of 1 minus x square minus x plus c and our answer to this question is we take minus sign common we will have cos inverse x into square root of 1 minus x square plus x plus c. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.