I'm Googling but not finding an answer. Are all calcuable definite integrals solvable using the method of Riemann sums? It seems like they should be. However, I have come across one that I can solve easily using known methods of integration. However, I am unable to solve it using (trapezoid) Riemann sums. I can post the specific problem, but really I'm just curious if anyone knows the answer to the question. Thanks!
P.S. It seems like, in order to solve using Riemann sums, you have to manipulate the summation into a few specific forms. So another way to phrase the question would be: is it always possible to manipulate the summation into some combination of those forms?
P.S.S. The integral that I couldn't get to work was $$\int_0^1{\frac{16x}{(4x^2+x)^2}dx}$$
One that I actually did get to work was $$\int_1^2{(x-1)^2dx}$$
Thanks!