 Hello and welcome to the session. In this session we discuss the following question which says evaluate integral e to the power x upon square root of 5 minus 4 e to the power x minus e to the power 2x dx. Before we move on to the solution let's recall one formula according to which we have integral of dx upon square root of a square minus x square is equal to sine inverse x upon a plus c where the c is the constant of integration. This is the key idea that we use in this question. Now we move on to the solution. We take let i be equal to the given integral that is integral e to the power x upon square root of 5 minus 4 into e to the power x minus e to the power 2x dx. We put e to the power x as t. Now differentiating both sides with respect to x we get e to the power x dx is equal to dt. Therefore i is equal to integral dt upon square root of 5 minus 4 t minus t square. Now further we get this is equal to integral dt upon square root of, now to make this expression in the square root a complete square we add 4 and subtract 4. So we get this is further equal to integral dt upon square root of 9 minus t plus 2 whole square. Further we get integral dt upon square root of 3 square minus t plus 2 whole square. By using this formula to evaluate the integral we get this is equal to sine inverse t plus 2 upon 3 plus c. That is a given integral i is equal to sine inverse of t plus 2 upon 3 plus c. Now putting the value of t as e to the power x we get i is equal to sine inverse of e to the power x plus 2 this whole upon 3 plus c. Thus integral e to the power x upon square root of 5 minus 4 into e to the power x minus e to the power 2x dx is equal to sine inverse of e to the power x plus 2 this whole upon 3 plus c. So this is the required solution. This completes the session hope you have understood the solution of this question.