 Hi and welcome to the session. I am Purva and I will help you with the following question. Evaluate the definite integral, integral limit from 1 to 4 mod x minus 1 plus mod x minus 2 plus mod x minus 3 dx. Let us now begin with the solution. Let us denote this definite integral by i. So we have i is equal to integral limit from 1 to 4 mod x minus 1 plus mod x minus 2 plus mod x minus 3 dx and let fx is equal to mod x minus 1 plus mod x minus 2 plus mod x minus 3. So we have fx is equal to minus x minus 1 minus x minus 2 minus x minus 3 if x is less than 1, right? fx is equal to x minus 1 minus x minus 2 minus x minus 3 if x is greater than equal to 1 and less than 2. fx is equal to x minus 1 plus x minus 2 minus x minus 3 if x is greater than equal to 2 and less than 3 and fx is equal to x minus 1 plus x minus 2 plus x minus 3 if x is greater than equal to 3. This implies fx is equal to now solving minus x minus 1 minus x minus 2 minus x minus 3 we get minus 3x plus 6 if x is less than 1. Solving x minus 1 minus x minus 2 minus x minus 3 we get fx is equal to minus x plus 4 if x is greater than equal to 1 and less than 2. Solving x minus 1 plus x minus 2 minus x minus 3 we get fx is equal to x if x is greater than equal to 2 and less than 3 and solving x minus 1 plus x minus 2 plus x minus 3 we get fx is equal to 3x minus 6 if x is greater than equal to 3. Therefore, we have i is equal to integral limit from 1 to 4 mod x minus 1 plus mod x minus 2 plus mod x minus 3 dx and this is equal to integral limit from 1 to 4 fx dx because we have taken fx is equal to mod x minus 1 plus mod x minus 2 plus mod x minus 3 and we can write this as this is equal to integral limit from 1 to 2 fx dx plus integral limit from 2 to 3 f of x dx plus integral limit from 3 to 4 f of x dx now this is equal to integral limit from 1 to 2 when x is between 1 and 2 we have f of x is equal to minus x plus 4 dx plus integral limit from 2 to 3 now when x is between 2 and 3 we have f of x is equal to x dx plus integral limit from 3 to 4 and when x is between 3 and 4 we have f of x is equal to 3x minus 6 dx this is equal to now integrating minus x plus 4 we get minus x square upon 2 plus 4x and limit is from 1 to 2 plus now integrating x we get x square upon 2 and limit is from 2 to 3 plus and now integrating 3x minus x we get 3x square upon 2 minus 6x and limit is from 3 to 4 this is equal to now putting the limits we get minus 4 upon 2 plus 8 minus minus 1 upon 2 plus 4 now here we have upper limit as 2 so putting 2 in place of x here we get minus 4 upon 2 plus 8 minus and we have lower limit as 1 so putting lower limit 1 in place of x here we get minus 1 upon 2 plus 4 plus again putting the limits here we get 9 upon 2 minus 4 upon 2 because we have upper limit as 3 so putting 3 in place of x here we get 9 upon 2 minus putting lower limit that is 2 in place of x here we get 4 upon 2 plus again putting here limits we get 3 into 16 upon 2 minus 6 into 4 minus 3 into 9 upon 2 minus 6 into 3 now here we have upper limit as 4 so putting 4 in place of x here we get 3 into 16 upon 2 minus 6 into 4 minus putting lower limit 3 in place of x here we get 3 into 9 upon 2 minus 6 into 3 here the common factor 2 will cancel out from denominator and numerator so we'll get here minus 2 so we get this is equal to minus 2 plus 8 is equal to 6 so 6 minus now minus 1 upon 2 plus 4 gives us 7 by 2 plus 9 by 2 minus 4 by 2 gives us 5 by 2 plus here again the common factor 2 will cancel out from numerator and denominator so we'll get 3 into 8 that is 24 minus 6 into 4 which is 24 minus 3 into 9 is 27 so 27 upon 2 minus 6 into 3 is 18 so 18 now this is equal to 6 minus now minus 7 by 2 plus 5 by 2 gives us minus 2 by 2 plus now 24 and 24 will cancel out so we get 0 minus 27 upon 2 minus 18 now here 2 and 2 will cancel out in numerator and denominator so we get 1 so we get this is equal to 6 minus 1 is 5 minus now 27 upon 2 minus 18 gives us minus 9 by 2 so we get this is equal to 5 plus 9 by 2 because minus into minus will become plus and this is equal to 19 by 2 so we get i is equal to 19 by 2 thus we get our answer as 19 upon 2 hope you have understood the solution bye and take care