The following questions are from the 2018 professional course test of numerical analysis of 武汉岩石所:

For the integral formula $int_{0}^{h} f(x) d x approx frac{h}{2}(f(0)+f(h))+alpha h^{2}left(f^{prime}(0)-f^{prime}(h)right)$:

**1.** We need to determine the value of $alpha$ to make it have the highest algebraic accuracy.

**2.** Suppose that the integrand `f(x)`

has the fourth order continuous derivative, it needs to be verified that when $alpha=frac{1}{12}$, the integral remainder of this formula is $R(f)=frac{1}{720} h^{5} f^{(4)}(eta), quad eta in(0, h)$.

```
Outer(Construct, {h/2 &, h/2 #1 &, h/2 #1^2 &, h/2 #1^3 &}, {0,
h}).{1, 1} +
Outer(Construct, {0 &, α h^2/2 &, α h^2/
2 2 #1 &, α h^2/2 3 #1^2 &}, {0, h}).{1, -1} ==
Integrate({1, x, x^2, x^3}, {x, 0, h})
```

From the results of the above code, I can know that when $alpha=frac{1}{6}$, the integral formula $int_{0}^{h} f(x) d x approx frac{h}{2}(f(0)+f(h))+alpha h^{2}left(f^{prime}(0)-f^{prime}(h)right)$ has the highest algebraic accuracy. But for the second question, I don’t know how to verify it with MMA. I want to get as many methods as possible to verify the second problem.