 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says integrate the following function. The function is x square into e raised to power x. Before starting with the solution, let us see the key idea behind the question. We integrate this function using integration by parts. That is integral of product of two functions that is integral fx gx dx is equal to fx into integral of gx dx minus integral of f dash x into integral gx dx the whole dx. Here we see that fx is the first function and gx is the second function and the choice of first function and second function is determined by the I let rule. This is the order of preference given to inverse function, logarithmic function, algebraic function, trigonometric function and exponential function. That is inverse function is given the highest preference and least preference given to exponential function for the first function. So let us start with the solution to this question. We have to integrate this function that means we have to find integral x square e raised to power x dx. Here we see that first function that is fx will be x square and second function that is gx will be e raised to power x because as per the I let rule whenever we have the product of algebraic and exponential function then algebraic function that is x square becomes the first function. Therefore we have this. Now on integration by parts we have let this be equal to I we call this integral I. So I can be written as first function into integral of second function. So x square into integral e raised to power x dx minus integration of derivative of first function into integral of second function. So minus integral d by dx of x square into integral e x dx. Now this is equal to x square e raised to power x because integral of e raised to power x is e raised to power x minus integral of d by dx of x square is 2x into integral of e raised to power x is e raised to power x dx. Since 2 is a constant so we take it out of the integral sign and we get minus 2 into integral x into e raised to power x dx. This can be further written as x square e raised to power x minus twice of integral x into e raised to power x dx. So what we do now is we again integrate this using by parts. So we will have x square into e raised to power x minus 2 into x into e raised to power x minus integral dx by dx is 1 into e raised to power x dx. Here we see this becomes a first function. This is a second function. So again using integration by parts we get first function into integral of second minus integration of d by dx of first function that is of x is 1 into integral of e raised to power x that is this. This is equal to x square into e raised to power x minus 2x into e raised to power x plus 2 into now again integral of e raised to power x dx is e raised to power x plus a constant c. Now if we take e raised to power x common from these terms we get e raised to power x into x square minus 2x plus 2 plus the constant. So our answer to this question is e raised to power x into x square minus 2x plus 2 plus c. So I hope that you understood the question and enjoyed the session. Have a good day.