{"id":2155,"date":"2022-02-22T15:11:07","date_gmt":"2022-02-22T06:11:07","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2155"},"modified":"2024-04-17T10:32:25","modified_gmt":"2024-04-17T01:32:25","slug":"%e9%83%a8%e5%88%86%e7%a9%8d%e5%88%86","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e9%83%a8%e5%88%86%e7%a9%8d%e5%88%86\/","title":{"rendered":"\u90e8\u5206\u7a4d\u5206"},"content":{"rendered":"

\u90e8\u5206\u7a4d\u5206\u306e\u30ad\u30e2<\/h3>\n

\u88ab\u7a4d\u5206\u95a2\u6570\u304c2\u3064\u306e\u95a2\u6570\u306e\u7a4d\u3067\u8868\u3055\u308c\u3066\u3044\u308b\u3068\u304d\uff0c\u3069\u3061\u3089\u304b\u7247\u65b9\u304c\u7c21\u5358\u306b\u7a4d\u5206\u3067\u304d\u3066\uff08\u305d\u3061\u3089\u3092 \\(f'(x)\\) \u3068\u3057\u3066\uff09\uff0c\u3069\u3061\u3089\u304b\u7247\u65b9\u306e\u5fae\u5206\u304c\u7c21\u5358\u306b\u3067\u304d\u308b\u5834\u5408\u306a\u3089\uff08\u305d\u3061\u3089\u3092 \\(g(x)\\) \u3068\u3057\u3066\uff09
\n$$\\int f'(x) g(x) \\,dx = f(x) g(x)\u00a0 -\\int f(x) g'(x)\\, dx$$<\/p>\n

\u5b9a\u7a4d\u5206\u306e\u5834\u5408\u306f\uff0c
\n$$\\int_a^b f'(x) g(x) \\,dx = \\Bigl[ f(x) g(x) \\Bigr]_a^b\u00a0 -\\int_a^b f(x) g'(x)\\, dx$$<\/p>\n

\u8a3c\u660e\u306f\uff0c
\n$$\\left( f(x) g(x) \\right)’ = f'(x) g(x) + f(x) g'(x)$$ \u306e\u4e21\u8fba\u3092\u7a4d\u5206\u3059\u308c\u3070
\n$$\\int \\left( f(x) g(x) \\right)’ \\,dx\u00a0 = f(x) g(x)= \\int f'(x) g(x)\\,dx + \\int f(x) g'(x) \\,dx$$
\n$$\\therefore\\ \\ \u00a0 \\int f'(x) g(x)\\,dx =\u00a0 f(x) g(x) -\\int f(x) g'(x) \\,dx$$<\/p>\n

\u4f8b\uff1a\\(\\displaystyle \\quad \\int \\log x\\, dx\\)<\/p>\n

\\begin{eqnarray}
\n\\int \\log x\\, dx &=& \\int 1\\cdot \\log x\\, dx \\\\
\n&=& \\int (x)’\\cdot \\log x\\, dx \\\\
\n&=& x \\log x -\\int x \\cdot(\\log x)’\\, dx\\\\
\n&=& x \\log x -\\int x \\cdot\\frac{1}{x}\\, dx\\\\
\n&=& x \\log x -\\int \\,dx \\\\
\n&=& x\\log x -x
\n\\end{eqnarray}<\/p>\n

\u53c2\u8003\uff1a\u9006\u4e09\u89d2\u95a2\u6570\u30fb\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u7a4d\u5206<\/h3>\n
\n
<\/div>\n
\n
\n

\u8cea\u554f\uff1a
\n\u9006\u4e09\u89d2\u95a2\u6570\u3084\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206\u306f\u3084\u3063\u305f\u3051\u3069\uff0c\u9006\u4e09\u89d2\u95a2\u6570\u3084\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u7a4d\u5206\u306f\u3069\u3046\u306a\u308b\u306e\uff1f<\/p>\n

\u56de\u7b54\uff1a
\n\u90e8\u5206\u7a4d\u5206\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u57fa\u672c\u7684\u306b\u9006\u4e09\u89d2\u95a2\u6570\u3084\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206\u304c\u308f\u304b\u308c\u3070\uff0c\u7a4d\u5206\u3082\u3067\u304d\u307e\u3059\u3002\u4ee5\u4e0b\u306b Maxima-Jupyter \u3067\u4e0d\u5b9a\u7a4d\u5206\u306e\u7b54\u3048\u3060\u3051\u66f8\u3044\u3066\u304a\u304d\u307e\u3059\u3002\u304c\u3093\u3070\u308c\uff01<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n
<\/div>\n
\n
\n

\u9006\u4e09\u89d2\u95a2\u6570\u306e\u7a4d\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
'<\/span>integrate<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n'<\/span>integrate<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n'<\/span>integrate<\/span>(<\/span>atan<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>atan<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}\\int {\\arcsin x}{\\;dx}=x\\,\\arcsin x+\\sqrt{1-x^2}\\]<\/div>\n<\/div>\n
\n
Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\int {\\arccos x}{\\;dx}=x\\,\\arccos x-\\sqrt{1-x^2}\\]<\/div>\n<\/div>\n
\n
Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}\\int {\\arctan x}{\\;dx}=x\\,\\arctan x-\\frac{\\log \\left(x^2+1\\right)}{2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u7a4d\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
'<\/span>integrate<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n'<\/span>integrate<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n'<\/span>integrate<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span> =<\/span> integrate<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}\\int {{\\rm asinh}\\; x}{\\;dx}=x\\,{\\rm asinh}\\; x-\\sqrt{x^2+1}\\]<\/div>\n<\/div>\n
\n
Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}\\int {{\\rm acosh}\\; x}{\\;dx}=x\\,{\\rm acosh}\\; x-\\sqrt{x^2-1}\\]<\/div>\n<\/div>\n
\n
Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\int {{\\rm atanh}\\; x}{\\;dx}=\\frac{\\log \\left(1-x^2\\right)}{2}+x\\,{\\rm atanh}\\; x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u90e8\u5206\u7a4d\u5206\u306e\u30ad\u30e2 <\/p>\n
\u88ab\u7a4d\u5206\u95a2\u6570\u304c2\u3064\u306e\u95a2\u6570\u306e\u7a4d\u3067\u8868\u3055\u308c\u3066\u3044\u308b\u3068\u304d\uff0c\u3069\u3061\u3089\u304b\u7247\u65b9\u304c\u7c21\u5358\u306b\u7a4d\u5206\u3067\u304d\u3066\uff08\u305d\u3061\u3089\u3092 \\(f'(x)\\) \u3068\u3057\u3066\uff09\uff0c\u3069\u3061\u3089\u304b\u7247\u65b9\u306e\u5fae\u5206\u304c\u7c21\u5358\u306b\u3067\u304d\u308b\u5834\u5408\u306a\u3089\uff08\u305d\u3061\u3089\u3092 \\(g(x)\\) \u3068\u3057\u3066\uff09 $$\\int f'(x) g(x) \\,dx = f(x) g(x)\u00a0 -\\int f(x) g'(x)\\, dx$$<\/p>\n
\u5b9a\u7a4d\u5206\u306e\u5834\u5408\u306f\uff0c $$\\int_a^b f'(x) g(x) \\,dx = \\Bigl[ f(x) g(x) \\Bigr]_a^b\u00a0 -\\int_a^b f(x) g'(x)\\, dx$$<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":24,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2155","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