# Nodes and Weights of Gauss-Chebyshev 2nd Calculator

## Calculates the nodes and weights of the Gauss-Chebyshev 2nd quadrature.

 $\normal(1)\ {\large\int_{\small -1}^{\hspace{25}\small 1}}\sqrt{1-x^2}f(x)dx\simeq {\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\[10](2)\ f(x)\rightarrow {\large \frac{g(x)}{\sqrt{1-x^2}}}\\\hspace{20}{\large\int_{\small -1}^{\hspace{25}\small 1}}g(x)dx\simeq {\large\sum_{\small i=1}^{n}}{\large \frac{w_i}{\sqrt{1-x_i^2}}}g(x_i)\\\hspace{5}nodes\hspace{35} x_i=-cos{\frac{(2i-1)\pi}{2n}}\\\hspace{5}weights\hspace{20} w_i={\large\frac{\pi}{n}}\\$
 order n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 n=2,3,4,..,100
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal Gaussian\ quadrature\\\hspace{10} {\large\int_{\small a}^{\hspace{25}\small b}}w(x)f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i), \ {\large\int_{\small a}^{\hspace{25}\small b}}g(x)dx\simeq{\large\sum_{\small i=1}^{n}}{\large\frac{w_{i}}{w(x_i)}}g(x_i)\\Gauss-Chebyshev\ 2nd\ quadrature\\\hspace{30} interval(a,b):\hspace{20} [-1,\ 1]\\\hspace{30} w(x):\hspace{80} \sqrt{1-x^2}\\\hspace{30} polynomialsl:\hspace{10} U_n (x)\\$

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