# Gauss-Lobatto integration Calculator

## Calculates the integral of the given function f(x) over the interval (a,b) using Gauss-Lobatto integration.

 $\normal{\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx\simeq w_{\small 1}f(-1)+w_nf(1)+{\large\sum_{\small i=2}^{n-1}}w_{i}f(x_i)\\$
f(x)
a
 , b
 partition 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
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 Generally, this integral can be rapidly and accurately calculated.The integrand f(x) is assumed to be analytic and non-periodic.$\normal Gauss-Lobatto\ integration\\[10](1)\ {\large\int_a^{\hspace{25}b}}f(x)dx={\large\frac{b-a}{2}\int_{\small -1}^{\hspace{25}\small 1}}f({\large\frac{b-a}{2}}y+{\large\frac{b+a}{2}})dy\\\hspace{190}x\rightarrow {\large\frac{b-a}{2}}y+{\large\frac{b+a}{2}}\\(2)\ \hspace{0}{\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx\simeq w_{\tiny 1}f({\small -1}){\small+}w_nf({\small1}){\small+}{\large\sum_{\small i=2}^{n-1}}w_{i}f(x_i)\\\hspace{30}nodes\hspace{35} x_i:\ P_{n-1}^{'}(x_i)=0\\\hspace{30}weights\hspace{20} w_{\small 1}=w_n={\large\frac{2}{n(n-1)}}\\\hspace{110} w_i={\large\frac{2}{n(n-1)[P_{n-1}(x_i)]^2}}\\$

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