 Hello and welcome to the session, let us discuss the following question. Question says using the properties of definite integrals evaluate the integral from minus 5 to 5 modulus of x plus 2 dx. First of all let us understand that definite integral from a to b fx dx is equal to definite integral from a to c fx dx plus definite integral from c to b fx dx. Always remember that c is any value between a and b. Now we will use this property of definite integrals as our key idea to solve the given question. Let us now start with the solution. We have to evaluate definite integral from minus 5 to 5 modulus of x plus 2 dx. Clearly we can see x plus 2 is greater than equal to 0 on closed interval minus 2 5 and x plus 2 is less than equal to 0 on closed interval minus 5 minus 2. Now given definite integral is equal to definite integral from minus 5 to minus 2 minus x plus 2 dx plus definite integral from minus 2 to 5 x plus 2 dx. Here we have used the property given in key idea. Now this definite integral is equal to minus x square upon 2 plus 2x and limits of the integral are from minus 5 to minus 2 plus now this integral is equal to x square upon 2 plus 2x and limits of the integral are from minus 2 to 5. Now substituting these limits in this function we get minus minus 2 square upon 2 plus 2 multiplied by minus 2 minus minus 5 square upon 2 plus 2 multiplied by minus 5. Now we will write this plus sign as it is substituting these limits in this function we get 5 square upon 2 plus 2 multiplied by 5 minus square of minus 2 upon 2 plus 2 multiplied by minus 2. Now this expression is further equal to minus 2 minus 4 solving this bracket we get 2 minus 4. Here we will write minus sign as it is now in this bracket we will write this minus sign as it is and solving these terms we get 25 upon 2 minus 10. Now here we will write this plus sign as it is and solving this bracket we get 25 upon 2 plus 10 minus 2 minus 4. Now simplifying further we get 2 plus 25 minus 20 upon 2 plus 25 plus 20 upon 2 minus minus 2. Now this is further equal to 2 plus 5 upon 2 plus 45 upon 2 plus 2. Now we know 2 plus 2 is equal to 4 and 5 upon 2 plus 45 upon 2 is equal to 50 upon 2. Now 50 upon 2 is equal to 25. So we can write this expression as 4 plus 25. Now adding these two terms we get 29. So we get given definite integral is equal to 29. So this is our required answer this completes the session hope you understood the solution take care and have a nice day.