 I am welcome to your session. Let us just ask the following question. The question says, integrate the following functions. The given function is 1 by x into square root of ax minus x square. Let us now begin with the solution. Let i is equal to integral of 1 by x into square root of ax minus x square with respect to x. Now quote, a by t in place of x, x equals to a by t implies dx is equal to minus a by t square dt. Now x into square root of ax minus x square will be equal to a by t as x is equal to a by t into square root of a into a by t minus a by t whole square and this is equal to a by t into square root of a square by t minus a square by t square and this is equal to a by t into square root of a square t minus a square by t square and this is equal to a square by t square into square root of t minus 1. So by putting a by t in place of x in this integral we get i equals to integral of 1 by a square by t square into square root of t minus 1 into minus a by t square dt and this is equal to integral of 1 by square root of t minus 1 into minus 1 by a into dt and this is equal to minus 1 by a into integral of 1 by square root of t minus 1 with respect to t. This is equal to minus 1 by a into integral of t minus 1 to the power minus 1 by 2 with respect to t. We know that integral of ax plus b to the power n with respect to x is equal to ax plus b to the power n plus 1 divided by derivative of ax plus b that is a plus c. So using this formula integral of t minus 1 to the power minus 1 by 2 with respect to t is equal to t minus 1 to the power minus 1 by 2 plus 1 by minus 1 by 2 plus 1 plus c and this is equal to minus 1 by a t minus 1 to the power 1 by 2 by 1 by 2 plus c and this is equal to minus 2 by a into square root of t minus 1 plus c. Now x is equal to a by t this implies t is equal to a by x so this is equal to minus 2 by a into square root of a by x minus 1 plus c this is equal to minus 2 by a into square root of a minus x by x plus c hence our required answer is minus 2 by a into square root of a minus x by x plus c this is our required answer. So this completes the session by and take care.