# Math Help - Improper Riemann intergal

1. ## Improper Riemann intergal

Here is a question we had on an assignment a while back, the damn thing was only worth 2 marks yet it took the guy the whole lecture to explain the answer. dumb. But anyway, here's the question and the solution.

What I want to know is whether there is an easier way to do this. I don't understand the N(b) bit in the solution and frankly for a 2 mark question it was far too easy to get half/zero marks considering the guy is the most stringent marker I've ever encountered and the solution is fairly long.

Show that,

$\lim_{b \to \infty} \int_0^b \frac{\sin(x)}{x} dx = \lim_{N \to \infty} \int_0^{\pi} \frac{\sin((N + 1/2)x)}{x} dx$.

I thought it would just be a simple, let $x = (N+1/2)y$ and go from there but apparently not...

Solution is in attachment.