Hi. Need some guidance to make the trapezoid rule more effective.

Let be a function. I want to find an approximation to the integral

.

I know that the trapezoid rule is given by

where .

(Notice that I want trapezoids/parts)

can be rewritten as

.

Ok, now over to the problem. To make the method more effective I use the approximation to integral from in in this way

.

The part that I dont understand in this new formula is why you just can divide ? The other part I do understand, because in you have already calculated where is even and , so in you only need to calculate with odd indexes .

I hope the problem is clear!? And I hope someone can help me explain and understand this formula. Maby it's a good idea to prove that it works? I dont know how, but maby a hint could help?