 Hi and welcome to our session. Let us discuss the following question. The question says integrate the following functions square root of 1 minus root x by 1 plus root x. Let us now begin with the solution. In this question, we have to integrate the function square root of 1 minus root x by 1 plus root x with respect to x. First put root x as cos t. Now root x equals to cos t implies x is equal to cos square t. This implies dx by dt is equal to 2 cos t into minus sin t. This implies dx is equal to minus 2 cos t sin t dt. Now we are going to substitute cos t in place of root x and minus 2 cos t sin t dt in place of dx plus integral root of minus square root of x by 1 plus root x dx is now equal to integral square root of 1 minus cos t by 1 plus cos t into minus 2 cos t sin t dt. Now this is equal to minus 2 into integral square root of we know the formula of 1 minus cos theta and 1 plus cos theta. So using this formula 1 minus cos t is equal to 2 sin square t by 2 and 1 plus cos t is equal to 2 cos square t by 2 into cos t. We know that sin theta is equal to 2 sin theta by 2 cos theta by 2 so sin t is equal to 2 sin t by 2 cos t by 2 dt. Now this is equal to minus 4 into integral square root of sin square t by 2 by cos square t by 2 into sin t by 2 cos t by 2 cos t dt. Now this is equal to minus 4 into integral sin t by 2 by cos t by 2 into sin t by 2 cos t by 2 cos t dt. Now cancel cos t by 2 from both numerator and denominator. So this is equal to minus 4 integral sin square t by 2 cos t dt and this is equal to minus 4 into integral sin square t by 2 is equal to minus, sorry 1 minus cos t by 2 into cos t dt. And this is equal to minus 2 into integral cos t dt plus 2 into integral cos square t dt. Now this is equal to minus 2 integral of cos t with respect to t is sin t plus 2 into integral cos square t is equal to 1 plus cos 2 t by 2 dt. Now this is equal to minus 2 sin t plus integral dt plus integral cos 2 t dt. And this is equal to minus 2 sin t plus t plus integral of cos 2 t with respect to t is sin 2 t by 2 plus c. Here c is denoting the sum of all the constants of these individuals. We know that cos t is equal to root x. This implies t is equal to cos inverse root x. We also know that cos square t plus sin square t is equal to 1. This implies sin t is equal to square root of 1 minus cos square t. Now here cos t is equal to root x. So sin t is equal to square root of 1 minus x. So now this is equal to minus 2 sin t is equal to square root of 1 minus x plus t is equal to cos inverse root x. Now sin 2 t is equal to 2 sin t cos t and sin 2 t by 2 is equal to 2 sin t cos t by 2 and this is equal to sin t cos t. Now sin t is equal to square root of 1 minus x and cos t is equal to root x. So we have square root of 1 minus x into square root of x plus c. Hence our required answer is minus 2 into square root of 1 minus x plus cos inverse root x plus root x into root 1 minus x plus c. So this time we end the session. Bye and take care.