 Hi and welcome to the session. Let us discuss the following question the question says Integrate the following functions given function is tan inverse square root of 1 minus x by 1 plus x Let's now begin with the solution in this question. We have to find integral of tan inverse Square root of 1 minus x by 1 plus x with respect to x We are going to put x as theta now x equals to cos theta implies dx by d theta is equal to minus sin theta This implies dx is equal to minus sin theta d theta so now But putting cos theta in place of x and minus sin theta d theta in place of dx we get Integral tan inverse Square root of 1 minus cos theta by 1 plus cos theta into minus sin theta d theta equal to integral tan inverse square root of 2 sin square theta by 2 by 2 cos square theta by 2 into minus sin theta d theta This is equal to integral tan inverse tan theta by 2 minus sin theta into d theta And this is equal to 1 by 2 integral theta into sin theta d theta We have minus sin here Now here we have product of two functions. So we are going to Integrate this by parts We will take the first function as the function which comes first in the word. I let now here theta is the algebraic function So we will treat theta as first function and sin theta as second function This is equal to minus 1 by 2 into theta into integral of sin theta With respect to theta minus integral derivative of theta into integral sin theta with respect to theta whole d theta This is equal to minus 1 by 2 into theta into minus cos theta minus integral 1 into minus cos theta d theta And this is equal to minus 1 by 2 minus theta cos theta plus sin theta plus c This is equal to theta cos theta by 2 minus sin theta by 2 plus c and this is equal to 1 by 2 theta cos theta minus 1 by 2 sin theta can be written as square root of 1 minus cos square theta as c This is equal to cos theta And this implies theta is equal to cos inverse x so we have 1 by 2 into cos inverse x into x minus 1 by 2 into square root of 1 minus x square plus c Hence our required answer is 1 by 2 into x cos inverse x minus square root of 1 minus x square plus c So this completes the session. Bye and take care