Numerical integration of $\int_0^2 \frac{1}{x+4}dx.$ I have homework problem. Determine the number of intervals required to approximate 
$$\int_0^2 \frac{1}{x+4}dx$$ to within $10^{-5}$ and computer the approximation using (a) Trapezoidal rule, (b) Simpson's rule, (c) Gaussian quadrature rule. I think the phrase "within $10^{-5}$,"means that the error term. 
I know that the m-point Newton-Cotes rule is defined by $$Q_{NC(m)}=\int_a^b p_{m-1}(x)dx,$$ where $p_{m-1}$ interpolates the function on $[a,b].$ So when $m=2,$ we call $Q_{NC(2)}$ trapezoidal rule, ans $Q_{NC(3)}$ is simpson's rule.
Can anyone explain what are these three rules and how I can proceed?? And what does $m$ represent?? Is it I am kind of lost in this class.. ans the text book is really really bad that I have no idea what it talks about...

It might also be asking you to use the remainder term formula.  I happen to remember that the remainder term for Simpson's rule using $n$ intervals (where here $n$ must be an even number) is $$ -\frac{(b-a)^5 f^{(4)}(\xi)}{180n^4}$$
where $a$ and $b$ are the limits of integration, $f$ is the integrand, and $\xi$ is between $a$ and $b$.  But this will only get you a lower bound for an $n$ that guarantees the error is less than $10^{-5}$.  There is a similar formula for the trapezoidal rule.
But I have absolutely no idea what it is for Gaussian quadrature!
