 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says integrate the following functions. The first question is x sin 3x. Before starting with the solution let us see the key idea behind the question. In this question we use integration by paths. We see that integral of product of two functions that is integral of fx gx dx is equal to fx into integral gx dx minus integral of f dash x into integral of gx dx the whole dx. Here f is the first function and gx is the second function. Now we use integration by paths in such questions where we have product of two functions. So one of the function acts as first function, second function acts as the second function and then we find out the integral of fx into gx dx. So we can say that integral of product of two functions is equal to first function into integral of second minus integral of differential coefficient of the first function into integral of the second function. Now the question is that if we are given x sin 3x. So how do we know which is the first function and which is the second function? So for that we have an islet rule that is inverse logarithmic algebraic trigonometric and exponential. So which function is need to be taken as the first function, which function is needed to be taken as second function is determined by the islet rule. For example in the product if we have an inverse function, so inverse function becomes the first function, if we have a logarithmic function that logarithmic function becomes the first function and so on. So this is the order of the preference for the first function. First preference would be inverse function, second preference would be logarithmic function, third would be algebraic, fourth would be trigonometric and last would be exponential. Again let us understand it more clearly. If we have an inverse function and a trigonometric function in the product then inverse function becomes the first function and trigonometric becomes the second. Similarly if we have logarithmic and algebraic then logarithmic becomes the first function and algebraic second and so on. So this is basically the order of preferences given to different functions while choosing the first function. So using this we find out the solution to this question. So let us start with the solution to this question. First of all what we do is we put fx to be equal to x and gx equal to sin 3x that is fx equal to x is the first function and gx equal to sin 3x is the second function. Now we see that according to the islet rule whenever we have an algebraic function that is x is an algebraic function and whenever we have a trigonometric function that is sin 3x is a trigonometric function. So algebraic function is given the preference to become the first function. So x will be the first function because that is an algebraic function. So we have this. Now integration by parts gives integration of x sin 3x dx will be equal to first function that is x into integral of second function that is sin 3x dx minus integral of d by dx of x into integral of sin 3x dx the whole into dx. Now this is equal to x into minus cos 3x y3 because integral of sin 3x is cos 3x divided by the differentiation of 3x that is 3. So here we have a minus sign because integral of sin 3x is minus cos 3x minus integral of now dx by dx is 1 into integral of sin 3x is again the same minus cos 3x by 3 into dx. This is equal to minus x cos 3x by 3 minus sin comes out of the integral we have plus integral of cos 3x into dx and 1 by 3 comes out of the integral sin. This is equal to minus x cos 3x divided by 3 plus 1 by 3 into sin 3x divided by 3 because integral cos 3x is sin 3x and we divided by 3 because differentiation of 3x is 3 plus a constant c because every time we find out integral we put a constant sign or constant number here and this is equal to minus x by 3 cos 3x plus 1 by 3 3 is on 9 sin 3x plus c. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.