\u305f\u3068\u3048\u3070<\/p>\n
$$f(x) = \\frac{2 x^3 + 3 x^2 -2 x -1}{x^2 + x -2}$$<\/p>\n
\u306e\u3088\u3046\u306b\uff0c\\(\\displaystyle \\frac{\u591a\u9805\u5f0f}{\u591a\u9805\u5f0f}\\) \u306e\u5f62\u306b\u306a\u3063\u3066\u3044\u308b\u95a2\u6570\u3092\u6709\u7406\u95a2\u6570<\/strong><\/span>\u3068\u3044\u3046\u3002<\/p>\n \u6709\u7406\u95a2\u6570\u3092\u7a4d\u5206\u3059\u308b\u969b\u306f\uff0c\u90e8\u5206\u5206\u6570\u306b\u5c55\u958b\u3057\u3066\u304b\u3089\u7a4d\u5206\u3059\u308b\u3002<\/p>\n \u4e0a\u8a18\u306e \\(f(x)\\) \u306e\u5834\u5408\u306f\uff0c\u5206\u5b50\u306e\u6b21\u6570\u304c\u5206\u6bcd\u3088\u308a\u9ad8\u3044\u306e\u3067\u307e\u305a\u5272\u308a\u7b97\u3057\u3066\u304b\u3089<\/strong>\u6b8b\u308a\u3092\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3<\/strong><\/span>\u3002<\/p>\n \\begin{eqnarray} \u3057\u305f\u304c\u3063\u3066<\/p>\n \\begin{eqnarray} \\begin{eqnarray} \u4e00\u6319\u306b \\(\\displaystyle \\int \\frac{dx}{1-x^2} = \\tanh^{-1} x + C\\) \u3068\u3057\u3066\u3082\u3088\u3044\u3060\u308d\u3046\u304c\uff0c$x$ \u306e\u7bc4\u56f2\u306b\u6ce8\u610f\u3002\\(\\tanh^{-1} x\\) \u306e\u5b9a\u7fa9\u57df\u306f $-1 < x < 1$\u3002<\/p>\n \u6709\u7406\u95a2\u6570\u306e\u7a4d\u5206\u3060\u304c\uff0c\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3059\u308c\u3070\u3088\u3044\u3068\u3044\u3046\u308f\u3051\u3067\u306f\u306a\u3044\u4f8b\u3002\u3053\u3093\u306a\u7a4d\u5206\uff0c\u3069\u3053\u3067\u4f7f\u3046\u3093\u3060\u3068\u601d\u3046\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u5b9f\u306f\u5b87\u5b99\u5e74\u9f62<\/a>\u306e\u3068\u3053\u308d\u3067\u4f7f\u3063\u305f\u308a\u3059\u308b\u306e\u3067\u3059\u3088\u3002<\/p>\n \u3053\u308c\u306f\u90e8\u5206\u5206\u6570\u3068\u304b\u3067\u306f\u306a\u304f\uff0c\u9006\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/a>\u3092\u601d\u3044\u51fa\u3057\u3066<\/p>\n $$\\int \\frac{dx}{1+x^2} = \\tan^{-1} x $$<\/p>\n \\begin{eqnarray} \u3059\u306a\u304a\u306b\u90e8\u5206\u5206\u6570\u5206\u89e3\u3059\u308b\u3068\uff0c<\/p>\n \\begin{eqnarray} \u3060\u304c…<\/p>\n \u3053\u308c\u306f\u4e0a\u8a182\u4f8b\u306e\u5408\u308f\u305b\u6280\u3067<\/p>\n \\begin{eqnarray} \u3053\u3093\u306a\u7a4d\u5206\uff0c\u3044\u3063\u305f\u3044\u3069\u3053\u3067\u4f7f\u3046\u3093\u3060\u3068\u601d\u3046\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\uff0c\u3053\u3093\u306a\u3068\u3053\u308d<\/a>\u3067\u4f7f\u308f\u308c\u3066\u3044\u308b\u3093\u3067\u3059\u3088\u3002<\/p>\n \\begin{eqnarray} \u6709\u7406\u95a2\u6570\u3068\u306f <\/p>\n \u305f\u3068\u3048\u3070<\/p>\n $$f(x) = \\frac{2 x^3 + 3 x^2 -2 x -1}{x^2 + x -2}$$<\/p>\n \u306e\u3088\u3046\u306b\uff0c\\(\\displaystyle \\frac{\u591a\u9805\u5f0f}{\u591a\u9805\u5f0f}\\) \u306e\u5f62\u306b\u306a\u3063\u3066\u3044\u308b\u95a2\u6570\u3092\u6709\u7406\u95a2\u6570\u3068\u3044\u3046\u3002<\/p>\u5272\u308a\u7b97\u5f8c\uff0c\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3057\u3066…<\/h3>\n
\nf(x) &=& \\frac{2 x^3 + 3 x^2 -2 x -1}{x^2 + x -2} \\\\
\n&=& \\frac{(2x + 1)(x^2 + x -2) + x+ 1}{x^2 + x -2}\\\\
\n&=& 2x + 1 + \\frac{ x+ 1}{(x+2)(x-1)}\\\\
\n&=& 2x + 1 + \\frac{1}{3} \\frac{1}{x+2} + \\frac{2}{3} \\frac{1}{x-1}
\n\\end{eqnarray}<\/p>\n
\n\\int f(x) dx &=& \\int (2 x + 1) dx + \\frac{1}{3} \\int \\frac{1}{x+2} dx + \\frac{2}{3} \\int \\frac{1}{x-1}dx\\\\
\n&=& x^2 + x + \\frac{1}{3} \\log |x+2| + \\frac{2}{3} \\log |x-1| + C\\\\
\n&=& x^2 + x + \\frac{1}{3} \\log \\left(|x+2| (x-1)^2 \\right) + C
\n\\end{eqnarray}<\/p>\n\u7c21\u5358\u306a\u4f8b<\/h3>\n
\n\\int \\frac{dx}{1-x^2} &=& \\int \\frac{dx}{(1+x)(1-x)} \\\\
\n&=& \\int \\left\\{\\frac{A}{1+x} + \\frac{B}{1-x} \\right\\} \\,dx \\\\
\n&=& \\frac{1}{2} \\int \\left\\{ \\frac{1}{1+x} + \\frac{1}{1-x}\\right\\} \\,dx \\\\
\n&=& \\frac{1}{2} \\left\\{ \\log |1+x| -\\log |1-x|\\right\\}\u00a0 + C\\\\
\n&=& \\frac{1}{2} \\log \\left| \\frac{1+x}{1-x} \\right| + C\\\\
\n&=& \\tanh^{-1} x + C
\n\\end{eqnarray}<\/p>\n\u3042\u307e\u308a\u898b\u304b\u3051\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u304c\u5c06\u6765\u4f7f\u3046\u4f8b<\/h3>\n
\u7df4\u7fd2\u554f\u984c 1. \\(\\displaystyle \\int \\frac{dx}{1+x^2} \\)<\/h4>\n
\u7df4\u7fd2\u554f\u984c 2. \\(\\displaystyle \\int \\frac{1-x^2}{(1+x^2)^2} dx\\)<\/h4>\n
\n\\int \\frac{1-x^2}{(1+x^2)^2} dx
\n&=& \\int \\frac{(1 + x^2) -2 x^2}{(1+x^2)^2}\u00a0\u00a0 dx \\\\
\n&=& \\int \\frac{x’ (1 + x^2) -x (1+ x^2)’}{(1+x^2)^2}\u00a0\u00a0 dx \\\\
\n&=& \\int \\frac{d}{dx}\\left\\{ \\frac{x}{1+x^2}\\right\\} dx \\\\
\n&=& \\frac{x}{1+x^2}
\n\\end{eqnarray}<\/p>\n
\n\\frac{1-x^2}{(1+x^2)^2} &=& \\frac{A}{1+x^2} + \\frac{B}{(1+x^2)^2} \\\\
\n&=& -\\frac{1}{1+x^2} + \\frac{2}{(1+x^2)^2}
\n\\end{eqnarray}<\/p>\n\u7df4\u7fd2\u554f\u984c 3. \\(\\displaystyle \\int \\frac{2 x^2}{(1+x^2)^2} dx\\)<\/h4>\n
\n\\int \\frac{2 x^2}{(1+x^2)^2} dx
\n&=& \\int \\left\\{ \\frac{1+x^2}{(1+x^2)^2} -\\frac{1-x^2}{(1+x^2)^2}\\right\\} dx \\\\
\n&=& \\int \\left\\{ \\frac{1}{1+x^2} -\\frac{1 -x^2}{(1+x^2)^2}\\right\\} dx \\\\
\n&=& \\tan^{-1} x -\\frac{x}{1+x^2}
\n\\end{eqnarray}<\/p>\n\u7df4\u7fd2\u554f\u984c 4. \\(\\displaystyle \\int \\frac{2}{(1+x^2)^2} dx\\)<\/h4>\n
\n\\int \\frac{2}{(1+x^2)^2} dx &=& \\int\\left\\{\\frac{2}{1+x^2} -\\frac{2x^2}{(1+x^2)^2} \\right\\}\\,dx\u00a0 \\\\
\n&=& 2 \\tan^{-1} x -\\left( \\tan^{-1} x -\\frac{x}{1+x^2}\\right) \\\\
\n&=& \\tan^{-1} x + \\frac{x}{1+x^2}
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"