{"id":566,"date":"2025-07-08T22:32:30","date_gmt":"2025-07-08T13:32:30","guid":{"rendered":"https:\/\/math-friend.com\/?p=566"},"modified":"2025-08-01T09:33:02","modified_gmt":"2025-08-01T00:33:02","slug":"%e3%80%90%e5%a4%a7%e9%98%aa%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e7%a9%8d%e5%88%86%e5%8c%ba%e9%96%93%e3%82%92%e5%b7%a7%e3%81%bf%e3%81%ab%e6%93%8d%e3%82%8b%e6%a5%b5%e9%99%90%e3%81%ae%e5%95%8f","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=566","title":{"rendered":"\u3010\u5927\u962a\u5927\u5b66\u5165\u8a66\u3011\u7a4d\u5206\u533a\u9593\u3092\u5de7\u307f\u306b\u64cd\u308b\u6975\u9650\u306e\u554f\u984c(2025)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\\(t>0\\)\u306e\u3068\u304d, \u4ee5\u4e0b\u3092\u793a\u305b.

(1)\\(\\displaystyle -\\frac{1}{t}<\\int_t^{2t}\\frac{\\sin{x}}{x^2}dx<\\frac{1}{t}\\)

(2) \\(\\displaystyle \\lim_{t\\rightarrow \\infty}\\int_t^{2t}\\frac{\\cos{x}}{x}dx=0\\)

(3) \\(\\displaystyle \\lim_{t\\rightarrow \\infty}\\int_1^{t}\\frac{\\sin{\\left(\\frac{3x}{2}\\right)}\\sin{\\left(\\frac{x}{2}\\right)}}{x}dx=\\frac{1}{2}\\int_1^2\\frac{\\cos{x}}{x}dx\\)
(2025 \u5927\u962a\u5927\u5b66\u7406\u7cfb[4])<\/span><\/p>\n\n\n\n

\u5b9a\u7a4d\u5206\u306e\u5927\u5c0f\u6bd4\u8f03, \u4e09\u89d2\u95a2\u6570\u306e\u7a4d\u548c\u516c\u5f0f, \u7a4d\u5206\u533a\u9593\u306e\u7d50\u5408, \u7a4d\u5206\u306e\u6975\u9650, \u306a\u3069\u3055\u307e\u3056\u307e\u306a\u30a8\u30c3\u30bb\u30f3\u30b9\u304c\u8a70\u307e\u3063\u305f\u826f\u554f\u3067\u3059. \u7279\u306b\u7a4d\u5206\u533a\u9593\u3092\u6a21\u7bc4\u89e3\u7b54\u306e\u3088\u3046\u306b\u5909\u63db\u3059\u308b\u554f\u984c\u306f\u79c1\u81ea\u8eab\u306f\u3042\u307e\u308a\u898b\u305f\u3053\u3068\u304c\u3042\u308a\u307e\u305b\u3093. \u7a4d\u5206\u533a\u9593\u306e\u7d50\u5408\u306b\u306f\u5b9a\u7a4d\u5206\u306e\u4ee5\u4e0b\u306e2\u6027\u8cea\u3092\u3046\u307e\u304f\u4f7f\u3063\u3066\u304f\u3060\u3055\u3044.
$$
\\begin{align}
\\int_a^bf(x)dx&=-\\int_b^af(x)dx\\\\
\\int_a^bf(x)dx+\\int_b^cf(x)dx&=\\int_a^cf(x)dx
\\end{align}
$$\u6700\u7d42\u7684\u306b\u793a\u3057\u305f\u3044\u306e\u306f3\u554f\u76ee\u306e\u7a4d\u5206\u306e\u6975\u9650\u306e\u7b49\u5f0f\u3067, \u524d2\u554f\u306f\u305d\u306e\u5c0e\u51fa\u3068\u306a\u3063\u3066\u3044\u307e\u3059.

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \u4eca\u8003\u3048\u3066\u3044\u308b\u7a4d\u5206\u533a\u9593\\([t, 2t]\\)\u306f\\(x=0\\)\u3092\u542b\u307e\u306a\u3044\u306e\u3067, \\(x^2>0\\)\u3067\u3042\u308b. \\(-1\\leq\\sin{x}\\leq 1\\)\u306e\u5168\u8fba\u3092\\(x^2(\\neq 0)\\)\u3067\u5272\u3063\u3066,
$$
-\\frac{1}{x^2}\\leq \\frac{\\sin{x}}{x^2}\\leq \\frac{1}{x^2}
$$
\u3092\u5f97\u308b. \u3053\u306e\u4e0d\u7b49\u5f0f\u306e\u5168\u8fba\u3092\\(x\\)\u3067\\([t,2t]\\)\u306e\u7bc4\u56f2\u3067\u7a4d\u5206\u3059\u308b\u3068,
$$
\\begin{align}
&-\\int_t^{2t}\\frac{1}{x^2}dx\\leq \\int_t^{2t}\\frac{\\sin{x}}{x^2}dx\\leq \\int_t^{2t}\\frac{1}{x^2}dx\\\\[1.5ex]
\\iff & \\left[\\frac{1}{x}\\right]_t^{2t}\\leq \\int_t^{2t}\\frac{\\sin{x}}{x^2}dx\\leq \\left[-\\frac{1}{x}\\right]_t^{2t}\\\\[1.5ex]
\\iff & -\\frac{1}{2t}\\leq \\int_t^{2t}\\frac{\\sin{x}}{x^2}dx\\leq \\frac{1}{2t}
\\end{align}
$$
\u3053\u3053\u3067, \\(\\frac{1}{2t}<\\frac{1}{t}\\)\u3067\u3042\u308b\u304b\u3089,
$$
-\\frac{1}{t}<\\left(-\\frac{1}{2t}\\leq\\right) \\int_t^{2t}\\frac{\\sin{x}}{x^2}dx\\left(\\leq \\frac{1}{2t}\\right)<\\frac{1}{t}
$$
\u3068\u306a\u308a, \u793a\u3055\u308c\u305f.<\/p>\n\n\n\n

(2) \u90e8\u5206\u7a4d\u5206\u304b\u3089,
$$
\\begin{align}
\\int_t^{2t}\\frac{\\cos{x}}{x}&=\\left[-\\frac{\\cos{x}}{x^2}\\right]_t^{2t}-\\int_t^{2t}\\frac{-\\sin{x}}{x^2}dx\\\\[1.5ex]
&=-\\frac{\\cos{(2t)}}{4t^2}+\\frac{\\cos{t}}{t^2}+\\int_t^{2t}\\frac{\\sin{x}}{x^2}dx
\\end{align}
$$
\u3053\u3053\u3067, \\(\\left|\\cos{(2t)}\\right|\\leq 1\\), \\(\\left|\\cos{(t)}\\right|\\leq 1\\)\u3067\u3042\u308b\u304b\u3089,
$$
\\begin{align}
\\left|-\\frac{\\cos{(2t)}}{4t^2}\\right|&=\\frac{\\left|\\cos{(2t)}\\right|}{\\left|4t^2\\right|}\\leq\\frac{1}{4t^2}\\to 0 \\,(t\\to \\infty)\\\\[1.5ex]
\\left|\\frac{\\cos{t}}{t^2}\\right|&=\\frac{\\left|\\cos{t}\\right|}{\\left|t^2\\right|}\\leq\\frac{1}{t^2}\\to 0 \\,(t\\to \\infty)
\\end{align}
$$\u3068\u306a\u308a,
$$
\\lim_{t\\rightarrow \\infty} \\frac{\\cos{(2t)}}{4t^2} = 0, \\,\\, \\lim_{t\\rightarrow \\infty} \\frac{\\cos{t}}{t^2} = 0
$$\u3067\u3042\u308b.

\u307e\u305f, (1)\u304b\u3089,
$$
\\left|\\int_t^{2t}\\frac{\\sin{x}}{x^2}dx\\right|<\\frac{1}{t}\\to 0 \\,(t\\to \\infty)
$$
\u3088\u308a,
$$
\\lim_{t\\rightarrow \\infty} \\int_t^{2t}\\frac{\\sin{x}}{x^2}dx = 0,
$$
\u3068\u306a\u308b. \u4ee5\u4e0a\u304b\u3089,
$$
\\begin{align}
\\lim_{t\\rightarrow \\infty}\\int_t^{2t}\\frac{\\cos{x}}{x}dx&=\\lim_{t\\rightarrow \\infty}\\left(-\\frac{\\cos{(2t)}}{4t^2}+\\frac{\\cos{t}}{t^2}+\\int_t^{2t}\\frac{\\sin{x}}{x^2}dx\\right)\\\\[1.5ex]
&=-\\lim_{t\\rightarrow \\infty}\\frac{\\cos{(2t)}}{4t^2}+\\lim_{t\\rightarrow \\infty}\\frac{\\cos{t}}{t^2}+\\lim_{t\\rightarrow \\infty}\\int_t^{2t}\\frac{\\sin{x}}{x^2}dx\\\\[1.5ex]
&=-0+0+0=0
\\end{align}
$$\u3068\u306a\u308a, \u793a\u3055\u308c\u305f.<\/p>\n\n\n\n

(3) \u7a4d\u548c\u516c\u5f0f\u3088\u308a,
$$
\\sin{\\left(\\frac{3x}{2}\\right)}\\sin{\\left(\\frac{x}{2}\\right)}=-\\frac{1}{2}\\left(\\cos{2x}-\\cos{x}\\right)
$$\u3067\u3042\u308b\u304b\u3089,
$$
\\int_1^t\\frac{\\sin{\\left(\\frac{3x}{2}\\right)}\\sin{\\left(\\frac{x}{2}\\right)}}{x}dx=
-\\frac{1}{2}\\int_1^t\\frac{\\cos{2x}}{x}dx+\\frac{1}{2}\\int_1^t\\frac{\\cos{x}}{x}dx
$$\u3068\u306a\u308b. \u3053\u3053\u3067, 1\u3064\u76ee\u306e\u7a4d\u5206\u3092\\(y=2x\\)\u3068\u3057\u3066\u7f6e\u63db\u7a4d\u5206\u3092\u884c\u3046. \u7a4d\u5206\u7bc4\u56f2\u306f,
$$
\\begin{array}{c|ccc}
x & 1 & \\rightarrow & t \\\\
\\hline
y & 2 & \\rightarrow & 2t \\\\
\\end{array}
$$\u3068\u306a\u308a, \\(dy=2dx\\)\u304b\u3089, \\(dx=\\frac{1}{2}dy\\)\u3067\u3042\u308b\u304b\u3089,
$$
\\begin{align}
\\int_1^t\\frac{\\cos{2x}}{x}dx&=\\int_2^{2t}\\frac{\\cos{y}}{\\frac{y}{2}}\\frac{1}{2}dy=\\int_2^{2t}\\frac{\\cos{y}}{y}dy\\\\[1.5ex]
=&\\int_2^{2t}\\frac{\\cos{x}}{x}dx
\\end{align}
$$\u3068\u306a\u308b(\u306a\u304a, \u5b9a\u7a4d\u5206\u306e\u5024\u306f\u7a4d\u5206\u5909\u6570\u306b\u7121\u95a2\u4fc2\u306e\u305f\u3081, \u6700\u5f8c\u306b\\(y\\)\u3092\u6539\u3081\u3066\\(x\\)\u3068\u7f6e\u304d\u76f4\u3057\u3066\u3044\u308b).

\u3053\u308c\u3092\u5143\u306e\u7a4d\u5206\u306b\u623b\u3059\u3068,
$$
\\begin{align}
&\\int_1^t\\frac{\\sin{\\left(\\frac{3x}{2}\\right)}\\sin{\\left(\\frac{x}{2}\\right)}}{x}dx=-\\frac{1}{2}\\int_2^{2t}\\frac{\\cos{x}}{x}dx+\\frac{1}{2}\\int_1^t\\frac{\\cos{x}}{x}dx\\\\[1.5ex]
&=\\frac{1}{2}\\left(\\int_
{2t}^{2}\\frac{\\cos{x}}{x}dx+\\int_1^t\\frac{\\cos{x}}{x}dx\\right)\\\\[1.5ex]
&=\\frac{1}{2}\\left(\\int_
{2t}^{2}\\frac{\\cos{x}}{x}dx+\\int_1^t\\frac{\\cos{x}}{x}dx+\\int_t^2\\frac{\\cos{x}}{x}dx-\\int_t^2\\frac{\\cos{x}}{x}dx\\right)\\\\[1.5ex]
&=\\frac{1}{2}\\left(\\int_
{2t}^{2}\\frac{\\cos{x}}{x}dx+\\int_2^t\\frac{\\cos{x}}{x}dx+\\int_1^t\\frac{\\cos{x}}{x}dx+\\int_t^2\\frac{\\cos{x}}{x}dx\\right)\\\\[1.5ex]
&=\\frac{1}{2}\\left(\\int_
{2t}^{t}\\frac{\\cos{x}}{x}dx+\\int_1^2\\frac{\\cos{x}}{x}dx\\right)\\\\[1.5ex]
&=\\frac{1}{2}\\left(-\\int_
{t}^{2t}\\frac{\\cos{x}}{x}dx+\\int_1^2\\frac{\\cos{x}}{x}dx\\right)
\\end{align}
$$
\u3053\u3053\u3067, \u7b2c\uff11\u9805\u76ee\u306e\u7a4d\u5206\u306f(2)\u3088\u308a,
$$
\\lim_{t\\rightarrow \\infty}\\int_t^{2t}\\frac{\\cos{x}}{x}dx=0
$$\u3067\u3042\u308a, \u7b2c2\u9805\u76ee\u306e\u7a4d\u5206\u306f\\(t\\)\u306b\u4f9d\u3089\u306a\u3044\u306e\u3067, \u5f53\u7136
$$
\\lim_{t\\rightarrow \\infty}\\int_1^2\\frac{\\cos{x}}{x}dx=\\int_1^2\\frac{\\cos{x}}{x}dx
$$\u3067\u3042\u308b.

\u3088\u3063\u3066,
$$
\\begin{align}
\\lim_{t \\to \\infty} \\int_1^t \\frac{\\sin\\left( \\frac{3x}{2} \\right) \\sin\\left( \\frac{x}{2} \\right)}{x} dx
&= \\frac{1}{2} \\left( -\\lim_{t \\to \\infty} \\int_t^{2t} \\frac{\\cos x}{x} dx + \\lim_{t \\to \\infty} \\int_1^2 \\frac{\\cos x}{x} dx \\right) \\\\[1.5ex]
&= \\frac{1}{2} \\left( -0 + \\int_1^2\\frac{\\cos x}{x} dx \\right)=\\frac{1}{2}\\int_1^2\\frac{\\cos x}{x} dx
\\end{align}
$$\u3068\u306a\u308a, \u793a\u3055\u308c\u305f.<\/p>\n<\/div><\/div>\n\n\n\n

(3)\u306f\u6700\u7d42\u7684\u306b\u793a\u3057\u305f\u3044\u53f3\u8fba\u306e\u5f62\u304c\u6c7a\u307e\u3063\u3066\u3044\u308b\u306e\u3067, \u305d\u3053\u306b\u5411\u304b\u3046\u3088\u3046\u8a66\u884c\u932f\u8aa4\u3057\u306a\u304c\u3089\u7a4d\u5206\u533a\u9593\u306e\u7d50\u5408\u3092\u884c\u306a\u3063\u3066\u304f\u3060\u3055\u3044. \u305d\u306e\u305f\u3081\u306b, \\(\\displaystyle \\int_t^2\\frac{\\cos{x}}{x}dx\\)\u3092\u8db3\u3057\u3066, \u5f15\u304f\u3068\u3044\u3046, \u5c11\u3057\u96e3\u3057\u3044\u5f0f\u5909\u5f62\u3092\u3057\u3066\u3044\u307e\u3059.

\u306a\u304a, (1)\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7a4d\u5206\u306e\u7d76\u5bfe\u5024\u3092\u8a55\u4fa1\u3057\u3066\u3082\u826f\u3044\u3067\u3057\u3087\u3046.
$$
\\begin{align}
\\left|\\int_t^{2t}\\frac{\\sin{x}}{x^2}dx\\right|&\\leq \\int_t^{2t}\\left|\\frac{\\sin{x}}{x^2}\\right|dx\\\\[1.5ex]
&\\leq \\int_t^{2t}\\frac{\\left|\\sin{x}\\right|}{\\left|x^2\\right|}dx\\\\[1.5ex]
& \\leq \\int_t^{2t}\\frac{1}{\\left|x^2\\right|}dx\\\\[1.5ex]
&= \\int_t^{2t}\\frac{1}{x^2}dx=\\frac{1}{2t}<\\frac{1}{t}\\,\\to 0\\,\\,(t\\to\\infty)
\\end{align}
$$

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