<\/p>\n
\\(\\displaystyle \\int_{-\\infty}^{\\infty} \\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} dx\\)<\/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<\/p>\n
\u307e\u305a\uff0c\u4e0d\u5b9a\u7a4d\u5206\u306f\u300c\u7121\u7406\u95a2\u6570\u306e\u7a4d\u5206\u300d\u306e\u7df4\u7fd2\u554f\u984c 4. \u305d\u306e\u3082\u306e\u3002<\/p>\n
\\(x = a \\tan\\theta\\) \u3068\u304a\u3044\u3066\u7f6e\u63db\u7a4d\u5206\u3002\\(\\displaystyle dx = \\frac{a}{\\cos^2\\theta}\\, d\\theta\\) \u3067\u3042\u308b\u304b\u3089…<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} dx &=&
\n\\int\\frac{1}{(a^2 + a^2 \\tan^2\\theta)^{\\frac{3}{2}}}\\frac{a}{\\cos^2\\theta} d\\theta \\\\
\n&=&
\n\\int\\frac{1}{a^3 (1 + \\tan^2\\theta)^{\\frac{3}{2}}}\\frac{a}{\\cos^2\\theta} d\\theta \\\\
\n&=& \\frac{1}{a^2} \\int \\cos\\theta\\, d\\theta \\\\
\n&=& \\frac{1}{a^2} \\sin\\theta \\\\
\n&=& \\frac{1}{a^2} \\tan\\theta \\times \\cos\\theta \\\\
\n&=& \\frac{1}{a^2} \\frac{\\tan\\theta}{\\sqrt{1 + \\tan^2\\theta}} \\\\
\n&=& \\frac{1}{a^2} \\frac{a \\tan\\theta}{\\sqrt{a^2 + a^2\\tan^2\\theta}} \\\\
\n&=& \\frac{1}{a^2} \\frac{x}{\\sqrt{a^2 + x^2}} \\\\
\n\\end{eqnarray}<\/p>\n
\u4e00\u65e6\u7b54\u3048\u304c\u308f\u304b\u308c\u3070\uff0c\u3044\u304b\u3088\u3046\u306b\u3082\u3053\u3058\u3064\u3051\u306f\u3067\u304d\u308b\u308f\u3051\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5225\u89e3\u3082\u53ef\u80fd\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} &=& \\frac{1}{\\sqrt{a^2 + x^2}} \\frac{1}{a^2 + x^2} \\\\
\n&=& \\frac{1}{\\sqrt{a^2 + x^2}} \\frac{1}{a^2} \\left( 1 -\\frac{x^2}{a^2 + x^2}\\right) \\\\
\n&=& \\frac{1}{a^2} \\left( \\frac{1}{\\sqrt{a^2 + x^2}} -x \\frac{x}{(a^2 + x^2)^{\\frac{3}{2}}}\\right) \\\\
\n&=& \\frac{1}{a^2} \\frac{d}{dx} \\frac{x}{\\sqrt{a^2 + x^2}} \\\\
\n\\therefore \\ \\ \\int \\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} dx &=& \\int \\frac{1}{a^2} \\frac{d}{dx} \\frac{x}{\\sqrt{a^2 + x^2}} dx \\\\
\n&=& \\frac{1}{a^2} \\frac{x}{\\sqrt{a^2 + x^2}}
\n\\end{eqnarray}<\/p>\n
\\(x = a \\tan\\theta\\) \u3068\u304a\u3044\u3066\u7f6e\u63db\u7a4d\u5206\u3060\u3068<\/p>\n
\\begin{eqnarray}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} dx &=&
\n\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}\\frac{1}{(a^2 + a^2 \\tan^2\\theta)^{\\frac{3}{2}}}\\frac{a}{\\cos^2\\theta} d\\theta \\\\
\n&=&
\n\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}\\frac{1}{a^3 (1 + \\tan^2\\theta)^{\\frac{3}{2}}}\\frac{a}{\\cos^2\\theta} d\\theta \\\\
\n&=& \\frac{1}{a^2} \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\cos\\theta\\, d\\theta \\\\
\n&=& \\frac{1}{a^2} \\Bigl[\\sin\\theta\\Bigr]_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\\\
\n&=&\\frac{1}{a^2} \\left(1 -\\left(-1\\right)\\right) \\\\
\n&=& \\frac{2}{a^2}
\n\\end{eqnarray}<\/p>\n
\u307e\u305f\u306f<\/p>\n
\\begin{eqnarray}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{(a^2 + x^2)^{\\frac{3}{2}}} dx
\n&=& \\lim_{b\\rightarrow \\infty}\\frac{1}{a^2} \\Bigl[\\frac{x}{\\sqrt{a^2 + x^2}}\\Bigr]_{-b}^b \\\\
\n&=& \\lim_{b\\rightarrow \\infty}\\frac{1}{a^2} \\Bigl[\\frac{x}{\\sqrt{x^2}}\\Bigr]_{-b}^b \\\\
\n&=& \\frac{1}{a^2}\\lim_{b\\rightarrow \\infty}\\left( \\frac{b}{|b|} -\\frac{-b}{|b|}\\right) \\\\
\n&=& \\frac{2}{a^2}
\n\\end{eqnarray}<\/p>\n
\\(\\displaystyle \\int_{-\\infty}^{\\infty} \\frac{1}{a^2 + x^2} dx\\)<\/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<\/p>\n
\u307e\u305a\u306f\u4e0d\u5b9a\u7a4d\u5206\u3002<\/p>\n
\\( x = a \\tan\\theta\\) \u3068\u304a\u3044\u3066\u7f6e\u63db\u7a4d\u5206<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{1}{a^2 + x^2} dx &=& \\int \\frac{1}{a^2} \\frac{1}{1+\\tan^2 \\theta} \\frac{a}{\\cos^2\\theta} \\,d\\theta \\\\
\n&=& \\frac{1}{a} \\int \\,d\\theta \\\\
\n&=& \\frac{\\theta}{a} \\\\
\n&=&\u00a0 \\frac{\\tan^{-1}\\frac{x}{a}}{a}
\n\\end{eqnarray}<\/p>\n
\u3042\u308b\u3044\u306f \\(\\displaystyle \\frac{d}{dx} \\tan^{-1} x = \\frac{1}{1+x^2}\\) \u304c\u308f\u304b\u3063\u3066\u3044\u308c\u3070\uff0c\u810a\u9ac4\u53cd\u5c04\u3067<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{1}{a^2 + x^2} dx &=& \\frac{1}{a} \\int \\frac{1}{1 + \\left(\\frac{x}{a}\\right)^2} d\\left(\\frac{x}{a} \\right)\\\\
\n&=& \\frac{\\tan^{-1}\\frac{x}{a}}{a}
\n\\end{eqnarray}<\/p>\n
\\( x = a \\tan\\theta\\) \u3068\u304a\u3044\u3066\u7f6e\u63db\u7a4d\u5206\u3059\u308c\u3070<\/p>\n
\\begin{eqnarray}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{a^2 + x^2} dx &=&
\n\\frac{1}{a} \\Bigl[\\theta\\Bigr]_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\\\
\n&=& \\frac{\\pi}{a}
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u3067\u5b89\u5fc3\u3057\u3066\u306f\u3044\u3051\u306a\u3044\u3002\u3053\u3053\u3067\u306f \\(a > 0\\) \u304c\u4eee\u5b9a\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u306b\u6ce8\u610f\u3002<\/p>\n
\\(a < 0\\) \u306e\u3068\u304d\u306b\u306f\uff0c
\n\\begin{eqnarray}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{a^2 + x^2} dx &=&
\n\\frac{1}{a} \\Bigl[\\theta\\Bigr]^{-\\frac{\\pi}{2}}_{\\frac{\\pi}{2}} \\\\
\n&=& -\\frac{\\pi}{a} \\\\
\n&=& \\frac{\\pi}{|a|}
\n\\end{eqnarray}<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3067 \\(a > 0\\) \u306e\u3068\u304d\u3082 \\(a < 0\\) \u306e\u3068\u304d\u306b\u3082\u4f7f\u3048\u308b\u5f62\u3067<\/p>\n
\\begin{eqnarray}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{a^2 + x^2} dx &=&
\n\\frac{\\pi}{|a|}
\n\\end{eqnarray}<\/p>\n
\u307e\u3068\u3081\u308b\u3068\uff0c\u5b9a\u7a4d\u5206\u306e\u5834\u5408\u306f\uff0c\u7a4d\u5206\u7bc4\u56f2\u306e\u5909\u63db\u304c\u3042\u308b\u304b\u3089\uff0c\u6700\u521d\u304b\u3089 \\( x = |a| \\tan\\theta\\) \u3068\u304a\u3044\u3066\u7f6e\u63db\u7a4d\u5206\u3059\u308c\u3070<\/p>\n
\\begin{eqnarray}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{a^2 + x^2} dx &=&
\n\\frac{\\pi}{|a|}
\n\\end{eqnarray}<\/p>\n
\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":3274,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6418","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t