有限区間（a,b）の数値積分

\( \normalsize Double\ exponential\ integration\\ (1)\ x\rightarrow \frac{b-a}{2}y+\frac{b+a}{2}\\ \hspace{30px} {\large\int_{\small a}^{\small b}}f(x)dx= {\large\int_{\small -1}^{\small 1}}f(\frac{b-a}{2}x+\frac{b+a}{2})\frac{b-a}{2}dx\\ (2) x\rightarrow \tanh(\frac{\pi}{2}\sinh(t))\\ \hspace{30px}{\large\int_{\small -1}^{\small 1}}f(x)dx={\large\int_{\small -\infty}^{\small \infty}}f(x(t))x'(t)dt\\ \hspace{140px}x(t)= \tanh(\frac{\pi}{2}\sinh(t))\\ \hspace{140px}x'(t)={\large\frac{\frac{\pi}{2}\cosh(t)}{\cosh^2(\frac{\pi}{2}\sinh(t))}}\\ (3)\ Trapezoid\\ \hspace{20px}S\simeq{\large\int_{-t_a}^{t_a}}f(x(t))x'(t)dt=h{\large\displaystyle \sum_{{\small j=-}\frac{N}{2}}^{\frac{N}{2}}}f(x(jh))x'(jh)\\ \hspace{240px}h={\large\frac{2t_a}{N}}\\ \)