\u4f8b\u3048\u3070\uff0c\\(\\displaystyle \\frac{(\\sin x)^2}{1 + \\cos x + 2 \\sin x}\\) \u306e\u3088\u3046\u306a\uff0c\\(\\sin x\\) \u3068 \\(\\cos x\\) \u306e\u6709\u7406\u95a2\u6570\u306e\u5f62\u306e\u95a2\u6570\u306e\u7a4d\u5206\u3002\\(\\displaystyle \\tan \\frac{x}{2} \\equiv t\\) \u3068\u3044\u3046\u5909\u63db\u3092\u3057\u3066\u7f6e\u63db\u7a4d\u5206\u3059\u308c\u3070\u3088\u3044\u3002<\/p>\n
<\/p>\n
\u00a0\uff08\u4e07\u7b56\u304c\u5c3d\u304d\u305f\u3089\u6700\u5f8c\u306e\u624b\u6bb5\u3068\u3057\u3066\uff09\\(\\displaystyle \\tan \\frac{x}{2} \\equiv t\\) \u3068\u3044\u3046\u5909\u63db\u3092\u3057\u3066\u7f6e\u63db\u7a4d\u5206\u3059\u308c\u3070\u3088\u3044\uff0c\u3068\u3044\u3046\u624b\u6cd5\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n
\u3053\u306e\u7f6e\u63db\u306b\u3088\u3063\u3066\uff0c\\(\\sin x\\)\uff0c\\(\\cos x\\)\uff0c\\(dx\\) \u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b \\(t\\) \u306e\u6709\u7406\u95a2\u6570\u3068\u306a\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\cos^2\\frac{x}{2} &=& \\frac{1}{1 + \\tan^2\\frac{x}{2}} = \\frac{1}{1+t^2} \\\\
\n\\sin^2 \\frac{x}{2} &=& 1 -\\cos^2\\frac{x}{2} = \\frac{t^2}{1 + t^2}
\n\\end{eqnarray}<\/p>\n
\u306a\u306e\u3067\uff0c\u3053\u308c\u3092\u4f7f\u3063\u3066<\/p>\n
\\begin{eqnarray}
\n\\sin x &=& 2 \\sin\\frac{x}{2} \\cos\\frac{x}{2} = 2 \\tan\\frac{x}{2} \\cos^2 \\frac{x}{2} \\\\
\n&=& \\frac{2t}{1+t^2} \\\\
\n\\cos x &=& \\cos^2\\frac{x}{2} -\\sin^2 \\frac{x}{2} = \\cos^2\\frac{x}{2} \\left(1 -\\tan^2 \\frac{x}{2} \\right) \\\\
\n&=& \\frac{1-t^2}{1+t^2}
\n\\end{eqnarray}<\/p>\n
\u307e\u305f\uff0c<\/p>\n
\\begin{eqnarray}
\ndt &=& d \\left(\\tan \\frac{x}{2} \\right) \\\\
\n&=& \\frac{1}{\\cos^2 \\frac{x}{2}} \\frac{dx}{2} \\\\
\n\\therefore\\ \\ dx&=& 2 \\cos^2 \\frac{x}{2} dt \\\\
\n&=& \\frac{2}{1+t^2} dt
\n\\end{eqnarray}<\/p>\n
\u306a\u306e\u3067\uff0c\u3053\u306e\u624b\u6cd5\u3067\u7f6e\u63db\u7a4d\u5206\u3092\u884c\u3046\u3068\uff0c\u7b54\u3048\u306f\u6700\u7d42\u7684\u306b\u306f \\(\\displaystyle t = \\tan \\frac{x}{2}\\) \u306e\u95a2\u6570\u3068\u3057\u3066\u66f8\u304b\u308c\u308b\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n
\u6700\u7d42\u7684\u306a\u7b54\u3048\u3092\uff0c\u534a\u89d2 \\(\\displaystyle\\frac{x}{2}\\) \u3067\u306f\u306a\u304f\uff0c\\(x\\) \u3092\u5f15\u6570\u3068\u3059\u308b\u4e09\u89d2\u95a2\u6570\u3067\u63cf\u304d\u76f4\u3057\u305f\u3044\u306a\u3041\u3068\u3044\u3046\u8981\u671b\u304c\u3042\u308b\u5834\u5408\u306b\u306f\uff0c\u4f8b\u3048\u3070\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f7f\u3063\u3066\u307f\u305f\u3089\u3069\u3046\u3060\u308d\u3046\u3002<\/p>\n
\\begin{eqnarray}
\nt = \\tan\\frac{x}{2} &=& \\frac{\\sin\\frac{x}{2}}{\\cos\\frac{x}{2}} \\\\
\n&=& \\frac{2 \\sin\\frac{x}{2} \\cos\\frac{x}{2}}{2 \\cos^2\\frac{x}{2}} \\\\
\n&=& \\frac{\\sin x}{1 + \\cos x} \\\\
\n&=& \\frac{ 1\u00a0 -\\cos x}{\\sin x}
\n\\end{eqnarray}<\/p>\n
\u4e07\u7b56\u3064\u304d\u3066\uff0c$\\tan \\frac{x}{2} = t$ \u3092\u4f7f\u3046\u3057\u304b\u306a\u3044\u304b\u306a\uff0c\u3068\u601d\u3046\u3068<\/p>\n
\\begin{eqnarray}
\n\\cos x &=& \\frac{1-t^2}{1+t^2} \\\\
\ndx &=& \\frac{2}{1+t^2} dt \\\\
\n\\therefore\\ \\ \\int \\frac{1}{\\cos x} \\,dx &=& \\int \\frac{1+t^2}{1-t^2}\\, \\frac{2}{1+t^2} \\,dt \\\\
\n&=& 2 \\int \\frac{1}{1-t^2}\\, dt \\\\
\n&=& \\int\\left(\\frac{1}{1-t} + \\frac{1}{1+t} \\right)\\, dt \\\\
\n&=& \\log |1 + t| -\\log |1-t| + C \\\\
\n&=& \\log\\left| \\frac{1+t}{1-t}\\right| + C \\\\
\n&=& \\log\\left| \\frac{1+\\tan\\frac{x}{2}}{1-\\tan\\frac{x}{2}}\\right| + C \\tag{1}
\n\\end{eqnarray}<\/p>\n
\u3068\u306a\u308b\u3002\u3057\u304b\u3057\uff0c\u5c11\u3057\u8003\u3048\u308b\u3068<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{1}{\\cos x} \\,dx &=& \\int \\frac{\\cos x}{\\cos^2 x} \\,dx \\\\
\n&=& \\int \\frac{\\cos x}{1-\\sin^2 x} \\,dx
\n\\end{eqnarray}<\/p>\n
\u3067\u3042\u308b\u304b\u3089\uff0c$u \\equiv \\sin x, \\ du = \\cos x \\, dx$ \u306e\u7f6e\u63db\u7a4d\u5206\u3067\u3067\u304d\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{1}{\\cos x} \\,dx &=& \\int \\frac{\\cos x}{1-\\sin^2 x} \\,dx \\\\
\n&=& \\int \\frac{1}{1-u^2} \\, du \\\\
\n&=& \\frac{1}{2} \\int \\left( \\frac{1}{1-u} + \\frac{1}{1+u}\\right) \\,du \\\\
\n&=& \\frac{1}{2}\u00a0 \\log\\left| \\frac{1+u}{1-u}\\right| + C \\\\
\n&=& \\frac{1}{2}\u00a0 \\log\\left| \\frac{1+\\sin x}{1-\\sin x}\\right| + C \\tag{2}
\n\\end{eqnarray}<\/p>\n
\u4eca\u5ea6\u306f\uff0c$(1)$ \u5f0f\u3068 $(2)$ \u5f0f\u306f\u679c\u305f\u3057\u3066\u540c\u3058\u3053\u3068\u3092\u8a00\u3063\u3066\u3044\u308b\u306e\u3060\u308d\u3046\u304b\uff1f\u3068\u3044\u3046\u7591\u554f\u304c\u51fa\u3066\u304f\u308b\u3060\u308d\u3046\u3002\u540c\u3058\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n
\u307e\u305a\uff0c\u3053\u3093\u306a\u7a4d\u5206\uff0c\u3044\u3063\u305f\u3044\u4f55\u306e\u5f79\u306b\u7acb\u3064\u306e\u304b\uff1f\u3068\u601d\u3046\u3067\u3057\u3087\uff1f \u3053\u306e\u7a4d\u5206\u306f\u96fb\u78c1\u6c17\u5b66\u306b\u304a\u3044\u3066\u8ef8\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u9759\u96fb\u5834\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u3093\u3067\u3059\u3088\u3002\u8a73\u7d30\u306f…<\/p>\n
\u3053\u308c\u306f $ u \\equiv \\cos\\theta$ \u3068\u3057\u305f\u7f6e\u63db\u7a4d\u5206\u3067…<\/p>\n
\u3053\u3093\u306a\u7a4d\u5206\uff0c\u3044\u3063\u305f\u3044\u4f55\u306e\u5f79\u306b\u7acb\u3064\u306e\u304b\uff1f\u3068\u601d\u3046\u3067\u3057\u3087\uff1f \u3053\u306e\u7a4d\u5206\u306f\u96fb\u78c1\u6c17\u5b66\u306b\u304a\u3044\u3066\u7403\u5bfe\u79f0\u306a\u96fb\u8377\u5206\u5e03\u306b\u3088\u308b\u9759\u96fb\u5834\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u5fc5\u8981\u306a\u3093\u3067\u3059\u3088\u3002\u8a73\u7d30\u306f…<\/p>\n
\\begin{eqnarray}
\n\\tan\\frac{x}{2} &\\equiv& t\\\\
\n\\sin x &=& \\frac{2 t}{1 + t^2} \\\\
\n\\cos x &=& \\frac{1 -t^2}{1 + t^2}
\n\\end{eqnarray}<\/p>\n
\u3053\u308c\u3089\u3092 $x$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3068<\/p>\n
$$x = 2 \\tan^{-1} t = \\sin^{-1} \\frac{2t}{1+t^2} = \\cos^{-1} \\frac{1 -t^2}{1+t^2}$$<\/p>\n
\u3053\u306e\u95a2\u4fc2\u306f\uff0c\u5b87\u5b99\u8ad6\u7684\u8ddd\u96e2\u306e\u3068\u3053\u308d\u3067\u3082\u4ee5\u5916\u306b\u6d3b\u7528\u3055\u308c\u3066\u3044\u305f\u308a\u3057\u307e\u3059<\/a>\u306e\u3067\uff0c\u30e1\u30e2\u3002<\/p>\n \u3053\u3093\u306a\u7a4d\u5206\uff0c\u4eba\u751f\u306e\u3069\u3053\u3067\u4f7f\u3046\u3093\u3060\uff1f\u3068\u601d\u3046\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\uff0c\u91cd\u529b\u5834\u4e2d\u306e\u5149\u306e\u7d4c\u8def<\/a>\u3092\u8abf\u3079\u308b\u3068\u304d\u306b\u5fc5\u8981\u306b\u306a\u308b\u3093\u3067\u3059\u3088\u3002<\/p>\n \\begin{eqnarray} \u4f8b\u3048\u3070\uff0c\\(\\displaystyle \\frac{(\\sin x)^2}{1 + \\cos x + 2 \\sin x}\\) \u306e\u3088\u3046\u306a\uff0c\\(\\sin x\\) \u3068 \\(\\cos x\\) \u306e\u6709\u7406\u95a2\u6570\u306e\u5f62\u306e\u95a2\u6570\u306e\u7a4d\u5206\u3002\\(\\displaystyle \\tan \\frac{x}{2} \\equiv t\\) \u3068\u3044\u3046\u5909\u63db\u3092\u3057\u3066\u7f6e\u63db\u7a4d\u5206\u3059\u308c\u3070\u3088\u3044\u3002<\/p>\u7df4\u7fd2\u554f\u984c 4. $\\displaystyle \\int \\frac{\\sin^3 x -1}{\\cos^2 x}\\, dx$<\/h3>\n
\n\\int \\frac{\\sin^3 x -1}{\\cos^2 x}\\, dx &=&
\n\\int \\frac{(1 -\\cos^2 x) \\sin x}{\\cos^2 x} \\, dx -\\int \\frac{1}{\\cos^2 x} \\, dx \\\\
\n&& \\qquad (\\cos x \\equiv t) \\\\
\n&=& \\int \\frac{t^2 -1}{t^2} \\, dt -\\tan x \\\\
\n&=& t + \\frac{1}{t} -\\tan x \\\\
\n&=& \\cos x + \\frac{1}{\\cos x} -\\tan x \\\\
\n&=& \\frac{\\cos^2 x + 1 -\\sin x}{\\cos x} \\\\
\n&=& \\frac{2\u00a0 -\\sin x -\\sin^2 x}{\\cos x}
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"