Tanh-Sinh integration (a,b) Calculator

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Calculates a table of the successive integral estimates of the given function f(x) over the interval (a,b) by doubling partitions from two to N using the Tanh-Sinh method.

This method is suitable for the function with endpoint singularities (±∞).
The integrand f(x) is assumed to be analytic and non-periodic.
It is calculated by increasing the number of partitions to double from 2 to N.

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

Tanh-Sinh integration (a,b) Calculator

Calculates a table of the successive integral estimates of the given function f(x) over the interval (a,b) by doubling partitions from two to N using the Tanh-Sinh method.

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Tanh-Sinh integration (a,b) Calculator

Calculates a table of the successive integral estimates of the given function f(x) over the interval (a,b) by doubling partitions from two to N using the Tanh-Sinh method.

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History

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