Exercise: Gauss-Chebyshev and an error bound

Compute $\int_{-1}^{1}\frac{e^x}{\sqrt{1-x^2}}\,dx$ with six guaranteed decimals using the error bound.

1. The Chebyshev error bound is bounded by e on (-1,1):

   $$|E|\le\frac{2\pi e}{2^{2n}(2n)!}$$

2. Testing n=1,2,3,4,5 gives the bounds:

   $$2.134933555,\ 0.044477782,\ 3.70648\cdot10^{-4},\ 1.65468\cdot10^{-6},\ 4.59633\cdot10^{-9}$$

3. The first $n$ that guarantees error below $10^{-6}$ is $n=5$.

   $$n=5$$

4. With the five Chebyshev nodes, the weighted sum gives:

   $$I\approx\frac{\pi}{5}\left(e^{-0.951057}+e^{-0.587785}+e^0+e^{0.587785}+e^{0.951057}\right)=3.977463$$