# Tanh-Sinh integration_2 (a,b) Calculator

## Calculates a table of the successive integral estimates of the given function f(x) over the interval (a,b) by increasing partitions from n to n+9 using Tanh-Sinh method.

 Interval (-∞, ∞) can approximate by (-ta, ta), since f(g(t))g'(t) converges rapidly to zero.Setup "auto" uses ta=(digit+1)^0.46. [example ta=3.5(14digit), 4.9(30digit)]It is calculated by increasing the number of partitions from n to n+9.
f(x)
a
 , b
 order n
ta
 auto manual -->
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt

 This method is suitable for the function with endpoint singularities (±∞).The integrand f(x) is assumed to be analytic and non-periodic.$\normal Tanh-Sinh\ integration\\\hspace{5}(1)\ x\rightarrow \frac{b-a}{2}y+\frac{b+a}{2}\\\hspace{30} {\large\int_{\small a}^{\hspace{25}\small b}}f(x)dx= {\large\int_{\small -1}^{\hspace{25}\small 1}}f(\frac{b-a}{2}y+\frac{b+a}{2})\frac{b-a}{2}dy\\[10]\hspace{20} y\rightarrow tanh(\frac{\pi}{2}sinh(t))\\\hspace{30}={\large\int_{\small -\infty}^{\hspace{25}\small \infty}}f(\frac{b-a}{2}y(t)+\frac{b+a}{2})y'(t)\frac{b-a}{2}dt\\\hspace{30}\simeq {\large\int_{\small -t_a}^{\hspace{25}\small t_a}}f(\frac{b-a}{2}y(t)+\frac{b+a}{2})y'(t)\frac{b-a}{2}dt\\\hspace{140}y(t)=tanh(\frac{\pi}{2}sinh(t))\\\hspace{140}y'(t)={\large\frac{\frac{\pi}{2}cosh(t)}{cosh^2(\frac{\pi}{2}sinh(t))}}\\[10](2)\ Trapezoid\\\hspace{20}S={\large\sum_{\small i=1}^{\small n}}f( \frac{b-a}{2}y_i+\frac{b+a}{2})w_i\\\hspace{10}nodes\\\hspace{20} y_i=tanh(\frac{\pi}{2}sinh(t_i)),\hspace{10}t_i=-t_a+(i-1)*h\ \\\hspace{10}weights\\\hspace{20} w_i={\large\frac{\frac{\pi}{2}cosh(t_i)}{cosh^2(\frac{\pi}{2}sinh(t_i))}}\frac{b-a}{2}h,\hspace{10}h={\large\frac{2t_a}{n-1}}\\$

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