 Hi and welcome to the session. Let's work out the following question. The question says evaluate Integral where limit goes from minus 5 to 0 fx dx Where fx is equal to mod x plus mod x plus 2 plus mod x plus 5 So let us start with the solution to this question You see that mod x is equal to minus x If x is less than 0 and it is equal to x If x is greater than equal to 0 mod x plus 2 is equal to minus x minus 2 If x is less than minus 2 and it is x plus 2 if x is greater than equal to minus 2 mod x plus 5 is equal to minus x minus 5 If x is less than minus 5 it is equal to x plus 5 If x is greater than equal to minus 5 therefore Integral where limit goes from minus 5 to 0 fx dx becomes equal to integral where limit goes from minus 5 to minus 2 fx dx Plus integral where limit goes from minus 2 to 0 fx dx That is equal to Integral where limit goes from minus 5 to minus 2. So when x lies between minus 5 and minus 2 fx is x plus 5 this minus x minus 2 minus x dx Plus integral where limit goes from minus 2 to 0 minus x plus x plus 2 plus x plus 5 dx This is equal to integral where limit goes from minus 5 to minus 2. Now this is 3 minus x dx Plus integral where limit goes from minus 2 to 0 x plus 7 dx This is equal to 3x Minus x square by 2 where limit goes from minus 5 to minus 2 Plus x square by 2 plus 7x where limit goes from minus 2 to 0 This is equal to 3 into minus 2 minus minus 2 the whole square by 2 minus 3 into minus 5 Plus minus 5 the whole square divided by 2 Plus Now when we put x to be equal to 0 this becomes 0. So we have minus sign here We have minus minus to the whole square by 2 plus 7 into minus 2 This is equal to minus 6 minus this will be 2 plus 15 plus 25 by 2 minus 2 minus 14 Here we will have plus 14 and this is equal to 63 by 2 which is our answer to this question. I hope that you understood the solution. I enjoyed this session. Have a good day