{"id":311,"date":"2025-07-04T03:09:59","date_gmt":"2025-07-03T18:09:59","guid":{"rendered":"https:\/\/math-friend.com\/?p=311"},"modified":"2025-08-01T09:26:31","modified_gmt":"2025-08-01T00:26:31","slug":"%e3%80%90%e4%b9%9d%e5%b7%9e%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91tan%e3%81%ae%e5%85%a5%e3%81%a3%e3%81%9f%e5%ae%9a%e7%a9%8d%e5%88%862025","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=311","title":{"rendered":"\u3010\u4e5d\u5dde\u5927\u5b66\u5165\u8a66\u3011tan\u306e\u5165\u3063\u305f\u5b9a\u7a4d\u5206(2025)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059. \u6570\u2162\u306e\u7bc4\u56f2\u306b\u306a\u308b\u306e\u3067, \u6587\u7cfb\u6570\u5b66\u306e\u7bc4\u56f2\u5916\u306e\u554f\u984c\u306b\u306a\u308a\u307e\u3059.<\/p>\n\n\n\n

(1) \u95a2\u6570\\(y=\\tan{x}\\)\u306b\u5bfe\u3057\u3066, \\(\\frac{dy}{dx}\\)\u3092\\(y\\)\u306e\u6574\u5f0f\u3067\u8868\u305b.
(2) \u4ee5\u4e0b\u306e\u5b9a\u7a4d\u5206\u3092\u6c42\u3081\u3088.
$$
\\int_0^{\\frac{\\pi}{4}}\\frac{\\tan^4{x}-\\tan^2{x}-2}{\\tan^2{x}-4}dx
$$
(2025 \u4e5d\u5dde\u5927\u5b66\u7406\u7cfb[2])<\/span><\/p>\n\n\n\n

\u3053\u306e\u554f\u984c\u306f(2)\u306e\u5b9a\u7a4d\u5206\u3092\u7f6e\u63db\u7a4d\u5206\u3067\u6c42\u3081\u308b\u305f\u3081\u306e\u8a98\u5c0e\u3068\u3057\u3066(1)\u304c\u3042\u308a\u307e\u3059. <\/p>\n\n\n\n

\n

(1) \\(y=\\tan{x}\\)\u306b\u3064\u3044\u3066, \u4e21\u8fba\u3092\\(x\\)\u3067\u5fae\u5206\u3059\u308b\u3068,
$$
\\frac{dy}{dx}=\\frac{1}{\\cos^2{x}}
$$
\u3053\u3053\u3067,
$$
1+\\tan^2{x}=\\frac{1}{\\cos^2{x}}
$$
\u3067\u3042\u308b\u304b\u3089,
$$
\\frac{dy}{dx}=1+\\tan^2{x}=1+y^2
$$
\u3092\u5f97\u308b.<\/p>\n\n\n\n

(2) \\(y=\\tan{x}\\)\u3068\u3057\u3066\u7f6e\u63db\u7a4d\u5206\u3092\u884c\u3046. \u7a4d\u5206\u7bc4\u56f2\u306f,
$$
\\begin{array}{c|ccc}
x & 0 & \\rightarrow & \\frac{\\pi}{4} \\\\
\\hline
y & 0 & \\rightarrow & 1 \\\\
\\end{array}
$$
\u3068\u306a\u308a, \u307e\u305f,
$$
\\frac{dy}{dx}=1+y^2
$$
\u304b\u3089,
$$
dx=\\frac{1}{1+y^2}dy
$$
\u3067\u3042\u308b. \u4ee5\u4e0a\u304b\u3089\u7f6e\u63db\u7a4d\u5206\u3092\u884c\u3046\u3068,
$$
\\begin{align}
&\\int_0^{\\frac{\\pi}{4}}\\frac{\\tan^4{x}-\\tan^2{x}-2}{\\tan^2{x}-4}dx=\\int_0^1\\frac{y^4-y^2-2}{y^2-4}\\cdot \\frac{1}{1+y^2}dy\\\\[1.5ex]
=&\\int_0^1\\frac{(y^2-2)(y^2+1)}{y^2-4}\\cdot \\frac{1}{1+y^2}dy=\\int_0^1\\frac{y^2-2}{y^2-4}dy
\\end{align}
$$
\u3053\u3053\u3067, \u591a\u9805\u5f0f\u306e\u5272\u308a\u7b97\u3068, \u90e8\u5206\u5206\u6570\u5206\u89e3\u306b\u3088\u308a,
$$
\\begin{align}
\\frac{y^2-2}{y^2-4}&=\\frac{(y^2-4)+2}{y^2-4}=1+\\frac{2}{y^2-4}\\\\[1.5ex]
&=1+\\frac{2}{(y-2)(y+2)}=1+2\\cdot\\frac{1}{4}\\left(\\frac{1}{y-2}-\\frac{1}{y+2}\\right)\\\\[1.5ex]
&=1+\\frac{1}{2}\\cdot\\frac{1}{y-2}-\\frac{1}{2}\\cdot\\frac{1}{y+2}
\\end{align}
$$
\u3068\u306a\u308b\u306e\u3067, \u7a4d\u5206\u306e\u7d9a\u304d\u3092\u884c\u3046\u3068,
$$
\\begin{align}
\\int_0^1\\frac{y^2-2}{y^2-4}dy&=\\int_0^1\\left(1+\\frac{1}{2}\\cdot\\frac{1}{y-2}-\\frac{1}{2}\\cdot\\frac{1}{y+2}\\right)dy\\\\[1.5ex]
&=\\left[y+\\frac{1}{2}\\log{|y-2|}-\\frac{1}{2}\\log{|y+2|}\\right]^1_0\\\\[1.5ex]
&=1-\\frac{1}{2}\\log{3}
\\end{align}
$$
\u3068\u8a08\u7b97\u3067\u304d\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

\u7f6e\u63db\u7a4d\u5206\u306e\u7f6e\u63db\u306b\u3064\u3044\u3066\u306f\u8a98\u5c0e\u306b\u4e57\u308c\u3070\u554f\u984c\u306a\u304f\u884c\u3048\u308b\u3067\u3057\u3087\u3046. \u305d\u306e\u5f8c\u306e\u6709\u7406\u95a2\u6570\u306e\u7a4d\u5206\u3001\u3064\u307e\u308a\u4e00\u822c\u306b\u6574\u5f0f\\(P(x)\\), \\(Q(x)\\)\u306b\u5bfe\u3057\u3066,
$$
\\int\\frac{Q(x)}{P(x)}dx
$$
\u306e\u7a4d\u5206\u306b\u95a2\u3057\u3066\u306f, 2\u3064\u306e\u5b9a\u77f3\u3092\u77e5\u3063\u3066\u304a\u304f\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059.

1\u3064\u3081\u306f, \u5206\u5b50\\(Q(x)\\)\u306e\u6b21\u6570\u304c\\(P(x)\\)\u306e\u6b21\u6570\u4ee5\u4e0a\u3067\u3042\u308b\u3068\u304d, \u6574\u5f0f\u306e\u5272\u308a\u7b97\u306b\u3088\u3063\u3066\u5206\u5b50\u306e\u6b21\u6570\u3092\u4e0b\u3052\u308b\u5fc5\u8981\u304c\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059. \u3053\u306e\u305f\u3081\u306b\u306f\u307e\u305a, \\(Q(x)\\)\u3092\\(P(x)\\)\u3067\u5272\u308a, \u305d\u306e\u5546\u3092\\(A(x)\\), \u4f59\u308a\u3092\\(R(x)\\) (\u305f\u3060\u3057, \\(R(x)\\)\u306e\u6b21\u6570\u306f\\(P(x)\\)\u306e\u6b21\u6570\u672a\u6e80)\u3068\u3059\u308b\u3068\u304d, \\(Q(x)\\)\u3092
$$
Q(x)=A(x)P(x)+R(x)
$$
\u306e\u5f62\u3067\u8868\u3057\u307e\u3059. \u3053\u306e\u4e21\u8fba\u3092\\(P(x)\\)\u3067\u5272\u308b\u3053\u3068\u3067,
$$
\\frac{Q(x)}{P(x)}=A(x)+\\frac{R(x)}{P(x)}
$$
\u3068\u306a\u308a, \\(R(x)\\)\u306e\u6b21\u6570\u306f\\(P(x)\\)\u306e\u6b21\u6570\u3088\u308a\u5c0f\u3055\u304f\u306a\u3063\u3066\u3044\u308b\u306e\u3067, \u3053\u306e\u5f62\u3067\u7a4d\u5206\u3092\u5b9f\u65bd\u3057\u307e\u3059.

2\u3064\u3081\u306f, \u90e8\u5206\u5206\u6570\u5206\u89e3\u3067\u3059. \\(\\frac{R(x)}{P(x)}\\)\u306e\u7a4d\u5206\u304c\u7c21\u5358\u306b\u3067\u304d\u308b\u5f62\u3067\u306a\u3044\u3068\u304d\u306f, \u4eca\u56de\u306e\u3088\u3046\u306b\u90e8\u5206\u5206\u6570\u5206\u89e3\u3092\u884c\u306a\u3063\u305f\u4e0a\u3067\u7a4d\u5206\u3092\u884c\u3044\u307e\u3059. \u90e8\u5206\u5206\u6570\u5206\u89e3\u3067\u3088\u304f\u4f7f\u308f\u308c\u308b\u5f62\u306f\u4ee5\u4e0b\u306a\u306e\u3067, \u3057\u3063\u304b\u308a\u899a\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046.
$$
\\begin{align}
\\frac{1}{x(x+1)}&=\\frac{1}{x}-\\frac{1}{x+1},\\\\[1.5ex]
\\frac{1}{x(x+a)}&=\\frac{1}{a}\\left(\\frac{1}{x}-\\frac{1}{x+a}\\right) \\,\\,(a\\neq 0),\\\\[1.5ex]
\\frac{1}{(x+a)(x+b)}&=\\frac{1}{b-a}\\left(\\frac{1}{x+a}-\\frac{1}{x+b}\\right) \\,\\,(a\\neq b).
\\end{align}
$$
\u90e8\u5206\u5206\u6570\u5206\u89e3\u3067\u5206\u6bcd\u30921\u6b21\u5f0f\u306b\u3067\u304d\u306a\u3044\u4f8b\u3082\u3042\u308a\u307e\u3059\u304c, \u4eca\u56de\u306e\u554f\u984c\u306b\u95a2\u3057\u3066\u306f\u4e0a\u306e\u90e8\u5206\u5206\u6570\u5206\u89e3\u3067\u5bfe\u5fdc\u3067\u304d\u307e\u3059.

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n