# 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.$\normal Tanh-Sinh\ integration\\\hspace{5}(1)\ x\rightarrow \frac{b-a}{2}y+\frac{b+a}{2}\\\hspace{30} {\large\int_{\small a}^{\hspace{25}\small b}}f(x)dx= {\large\int_{\small -1}^{\hspace{25}\small 1}}f(\frac{b-a}{2}y+\frac{b+a}{2})\frac{b-a}{2}dy\\[10]\hspace{20} y\rightarrow tanh(\frac{\pi}{2}sinh(t))\\\hspace{30}={\large\int_{\small -\infty}^{\hspace{25}\small \infty}}f(\frac{b-a}{2}y(t)+\frac{b+a}{2})y'(t)\frac{b-a}{2}dt\\\hspace{30}\simeq {\large\int_{\small -t_a}^{\hspace{25}\small t_a}}f(\frac{b-a}{2}y(t)+\frac{b+a}{2})y'(t)\frac{b-a}{2}dt\\\hspace{140}y(t)=tanh(\frac{\pi}{2}sinh(t))\\\hspace{140}y'(t)={\large\frac{\frac{\pi}{2}cosh(t)}{cosh^2(\frac{\pi}{2}sinh(t))}}\\[10](2)\ Trapezoid\\\hspace{20}S={\large\sum_{\small i=1}^{\small n}}f( \frac{b-a}{2}y_i+\frac{b+a}{2})w_i\\\hspace{10}nodes\\\hspace{20} y_i=tanh(\frac{\pi}{2}sinh(t_i)),\hspace{10}t_i=-t_a+(i-1)*h\ \\\hspace{10}weights\\\hspace{20} w_i={\large\frac{\frac{\pi}{2}cosh(t_i)}{cosh^2(\frac{\pi}{2}sinh(t_i))}}\frac{b-a}{2}h,\hspace{10}h={\large\frac{2t_a}{n-1}}\\$

Sending completion

To improve this 'Tanh-Sinh integration_2 (a,b) Calculator', please fill in questionnaire.
Male or Female ?
Male Female
Age
Under 20 years old 20 years old level 30 years old level
40 years old level 50 years old level 60 years old level or over
Occupation
Elementary school/ Junior high-school student
High-school/ University/ Grad student A homemaker An office worker / A public employee
Self-employed people An engineer A teacher / A researcher
A retired people Others
Useful?
Very Useful A little Not at All
Purpose of use?