{"id":3274,"date":"2022-07-15T16:46:31","date_gmt":"2022-07-15T07:46:31","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=3274"},"modified":"2024-04-17T10:35:37","modified_gmt":"2024-04-17T01:35:37","slug":"%e5%ba%83%e7%be%a9%e3%81%ae%e7%a9%8d%e5%88%86","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e5%ba%83%e7%be%a9%e3%81%ae%e7%a9%8d%e5%88%86\/","title":{"rendered":"\u5e83\u7fa9\u306e\u7a4d\u5206\u30fb\u7121\u9650\u7a4d\u5206"},"content":{"rendered":"

\\(\\displaystyle \\int_a^b f(x) \\,dx\\) \u306b\u304a\u3044\u3066\uff0c\u88ab\u7a4d\u5206\u95a2\u6570 \\(f(x)\\) \u304c \\([a, b]\\) \u3067\u6709\u754c\u3067\u306a\u304b\u3063\u305f\u308a\u9023\u7d9a\u3067\u306a\u304b\u3063\u305f\u308a\u3059\u308b\u5834\u5408<\/strong><\/span>\u3084\uff0c\\(a \\rightarrow -\\infty\\) \u3042\u308b\u3044\u306f \\(b \\rightarrow \\infty\\) \u306e\u3088\u3046\u306a\u7121\u9650\u7a4d\u5206\u306e\u5834\u5408<\/strong><\/span>\u3002<\/p>\n

\\(\\displaystyle \\int_a^b f(x) \\,dx\\) \u306b\u304a\u3044\u3066\uff0c\u88ab\u7a4d\u5206\u95a2\u6570 \\(f(x)\\) \u304c\u4f8b\u3048\u3070 \\(x = a\\) \u3067\u6709\u754c\u3067\u306f\u306a\u304b\u3063\u305f\u308a\uff0c\u9023\u7d9a\u3067\u306f\u306a\u304b\u3063\u305f\u308a\u3059\u308b\u5834\u5408\u3002<\/p>\n

\u305f\u3068\u3048\u3070\uff0c \\(\\displaystyle f(x) = x^{-p} = \\frac{1}{x^p}, \\ 0 < p < 1\\) \u306e\u5834\u5408\u3002\\(x = 0\\) \u3067 \\(f(x)\\) \u306e\u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u308b\u306e\u3067\uff0c\u56f0\u3063\u3066\u3057\u307e\u3044\u305d\u3046\u306b\u306a\u308b\u304c\uff0c\\(1 -p > 0\\) \u3067\u3042\u308b\u304b\u3089\uff0c\\(0^{1-p}= 0\\) \u306a\u306e\u3067<\/p>\n

\\begin{eqnarray}
\n\\int_0^2 f(x) dx &=& \\int_0^2 x^{-p}\\,dx \\\\
\n&=& \\frac{1}{1-p} \\left[ f^{1-p} \\right]_0^2 \\\\
\n&=& \\frac{2^{1-p} -0^{1-p}}{1-p} \\\\
\n&=& \\frac{2^{1-p} }{1-p}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c\u666e\u901a\u306b\u5b9a\u7a4d\u5206\u3067\u304d\u308b\u3002\u88ab\u7a4d\u5206\u95a2\u6570 \\(f(x)\\) \u304c \\(x = 0\\) \u3067\u6709\u754c\uff08\u6709\u9650\u306e\u5024\u3092\u6301\u3064\u3053\u3068\uff09\u3067\u306a\u304f\u3066\u3082\uff0c\u539f\u59cb\u95a2\u6570 \\(\\displaystyle F(x) = \\int^x f(u)\\, du\\) \u304c \\(x = 0\\) \u3067\u6709\u754c\u3067\u3042\u308c\u3070\uff0c\u7279\u306b\u554f\u984c\u306a\u304f\u5b9a\u7a4d\u5206\u3067\u304d\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002<\/p>\n

\u307e\u305f\uff0c\u7a4d\u5206\u533a\u9593\u304c\u7121\u9650\u3068\u306a\u308b\u7a4d\u5206\uff08\u7121\u9650\u7a4d\u5206\uff09\u306b\u3064\u3044\u3066\u306f\uff0c<\/p>\n

$$ \\int_a^{\\infty} f(x) \\,dx = \\left[F(x)\\right]_a^{\\infty} = F(\\infty) -F(a)$$<\/p>\n

\u306a\u3069\u3068\uff0c\u76f4\u63a5 \\(\\infty\\) \u3092\u5165\u308c\u3066 \\(F(\\infty)\\) \u3068\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u306a\u3044\uff08\\(\\infty\\) \u306f\u6570\u5024\u3067\u306f\u306a\u3044\uff09\u306e\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6975\u9650\u3092\u4f7f\u3046\u3002<\/p>\n

$$ \\int_a^{\\infty} f(x) \\,dx = \\lim_{b \\rightarrow \\infty} \\int_a^b f(x)\\,dx = \\lim_{b \\rightarrow \\infty} F(b) -F(a)$$<\/p>\n

$$ \\int_{-\\infty}^{\\infty} f(x) \\,dx = \\lim_{a \\rightarrow -\\infty} \\lim_{b \\rightarrow \\infty} \\int_a^b f(x)\\,dx=\\lim_{b \\rightarrow \\infty} F(b) -\\lim_{a \\rightarrow -\\infty}F(a)$$<\/p>\n

\u7c21\u5358\u306a\u4f8b\uff1a<\/p>\n

\\begin{eqnarray}
\n\\int_0^{\\infty} e^{-x} \\,dx &=&\\lim_{b \\rightarrow \\infty} \\int_0^{b} e^{-x} \\,dx\\\\
\n&=& \\lim_{b \\rightarrow \\infty} (-e^{-b}) -(-e^0)\\\\
\n&=& 0 -(-1) = 1
\n\\end{eqnarray}<\/p>\n

\u4e00\u898b\uff0c\u7121\u9650\u533a\u9593\u306e\u7a4d\u5206\u3060\u304c\uff0c\\(\\tan\\theta\\) \u3092\u542b\u3080\u5909\u6570\u5909\u63db\u306b\u3088\u308b\u7f6e\u63db\u7a4d\u5206\u306b\u3059\u308c\u3070\uff0c\\(\\displaystyle -\\frac{\\pi}{2} \\le \\theta \\le \\frac{\\pi}{2}\\) \u306e\u6709\u9650\u533a\u9593\u3067\u306e\u5b9a\u7a4d\u5206\u306b\u5e30\u7740\u3059\u308b\u5834\u5408\u3082\u3042\u308b\u3002\u4ee5\u4e0b\u306e\u7df4\u7fd2\u554f\u984c\u3092\u53c2\u7167\u3002<\/p>\n

\u7df4\u7fd2\u554f\u984c 1. $\\displaystyle \\int_{-\\infty}^{\\infty} \\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} dx$<\/h4>\n

\u3053\u3093\u306a\u7a4d\u5206\uff0c\u3044\u3063\u305f\u3044\u3069\u3053\u3067\u4f7f\u3046\u3093\u3060\u3068\u601d\u3046\u3067\u3057\u3087\u3046\u304c\uff0c1\u5e74\u751f\u5f8c\u671f\u306e\u96fb\u78c1\u6c17\u5b66I\u3067\uff0c\u4e00\u69d8\u306a\u7dda\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834<\/a>\u3092\u6c42\u3081\u308b\u969b\u306b\u4f7f\u3046\u3093\u3067\u3059\u3088\uff01\u3000\u8a73\u7d30\u306f…<\/p>\n