Grading some exams I noticed that the following is true: $$\int_1^3 (3x+1)(x-1)dx = \int_1^3(3x+1)dx +\int_1^3(x-1)dx.$$ Of course in general $$\int_a^b f(x)g(x)dx \neq \int_a^bf(x)dx+\int_a^bg(x)dx.$$ But is there a general statement of which the equality statement above is an example or is it truly just one of those weird coincidences?
Asked
Active
Viewed 60 times
4
-
Truly a coincidence. – uniquesolution Dec 01 '23 at 20:26
-
this only holds for special f and g, where if the sum of the independently integrated functions is equated to the base integral, when evaluated at some point [0,n] , the factorization of the equality gives the values of n to be equal to points of [a,b] . – Amy Skinner Dec 01 '23 at 20:41
-
1While it is a coincidence, it is not a particularly weird one. When one writes down "nice" problems for exam questions, one tends to choose low integers and obtain low integer solutions. This means that there are only a fairly small set of numbers we are working from, and it is not at all uncommon that the same number will arise from different sources. – Paul Sinclair Dec 03 '23 at 00:04