半無限領域（a,∞）の数値積分

\( \normalsize Double\ exponential\ integration\\ (1)\ x\rightarrow e^{{\normalsize\frac{\pi}{2}}\sinh(t)}\\ \ {\large\int_{\tiny a}^{\tiny\infty}}f(x)dx={\large\int_{\tiny 0}^{\tiny\infty}}f(x+a)dx={\large\int_{\tiny -\infty}^{\tiny\infty}}f(x(t)+a)x'(t)dt\\ \hspace{140px}x(t)=e^{{\normalsize\frac{\pi}{2}}\sinh(t)}\\ \hspace{140px}x'(t)=e^{{\normalsize\frac{\pi}{2}}\sinh(t)}\frac{\pi}{2}\cosh(t)\\ (2)\ Trapezoid\\ \hspace{20px}S\simeq{\large\int_{-t_a}^{t_a}}f(x(t))x'(t)dt=h{\large\displaystyle \sum_{{\small j=-}\frac{N}{2}}^{\frac{N}{2}}}f(x(jh))x'(jh)\\ \hspace{240px}h={\large\frac{2t_a}{N}}\\ \)