# Tanh-Sinhlϕ̕_Əd

## Tanh-Sinhlϕ̕_inodesjƏd݁iweightsjvZ܂B

 $\normal{\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx\simeq{\large\int_{-t_a}^{\hspace{25}t_a}}f(x(t))x'(t)dt\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\\ nodes\\\hspace{20} x_i=tanh(\frac{\pi}{2}sinh(t_i)),\hspace{10}t_i=-t_a+(i-1)h\\[10]\ weights\\\hspace{20} w_i={\large\frac{\frac{\pi}{2}cosh(t_i)}{cosh^2(\frac{\pi}{2}sinh(t_i))}}h,\hspace{10}h={\large\frac{2t_a}{n-1}}\\\vspace{10}$
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 $\normal Tanh-Sinh\ integration\\\hspace{5}(1)\hspace{1} {\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx={\large\int_{\small -\infty}^{\hspace{25}\small \infty}}f(x(t))x'(t)dt\\\hspace{90}\simeq{\large\int_{-t_a}^{\hspace{25}t_a}}f(x(t))x'(t)dt\\\hspace{140}x(t)=tanh(\frac{\pi}{2}sinh(t))\\\hspace{140}x'(t)={\large\frac{\frac{\pi}{2}cosh(t)}{cosh^2(\frac{\pi}{2}sinh(t))}}\\(2)\ Trapezoid\\\hspace{20} {\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\\hspace{10}nodes\\\hspace{20} x_i=tanh(\frac{\pi}{2}sinh(t_i)),\hspace{10}t_i=-t_a+(i-1)h\\[10]\hspace{10}weights\\\hspace{20} w_i={\large\frac{\frac{\pi}{2}cosh(t_i)}{cosh^2(\frac{\pi}{2}sinh(t_i))}}h,\hspace{10}h={\large\frac{2t_a}{n-1}}\\$

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