Was trying to use chebyshev polynomial to obtain a cubic approximation to $f(x)=\frac{1}{x} $
I did it over the interval $[-1,1] $
Solving gave me the following four definite integrals:
$C_0=\frac{2}{\pi}\int_{-1}^1\frac{1}{x\sqrt{1-x^2}}dx$
$C_1=\frac{2}{\pi}\int_{-1}^1\frac{1}{\sqrt{1-x^2}}dx$
$C_2=\frac{2}{\pi}\int_{-1}^1\frac{2x^2-1}{x\sqrt{1-x^2}}dx$
$C_3=\frac{2}{\pi}\int_{-1}^1\frac{4x^3-3x}{x\sqrt{1-x^2}}dx$
Now my problem is how to integrate these integrals analytically without resorting to numerical methods such as trapezoidal rule or Simpson rule.
Of course there's an exception: $C_1$ & $C_3$
if you look closer you see that all of them are related to each other and that just $x = sin(\theta)$ solves them all.