{"id":32657,"date":"2023-04-04T13:00:16","date_gmt":"2023-04-04T13:00:16","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32657"},"modified":"2025-03-27T12:11:50","modified_gmt":"2025-03-27T12:11:50","slug":"%e4%b8%8d%e5%ae%9a%e7%a7%af%e5%88%86%e4%b8%8e%e5%9f%ba%e6%9c%ac%e7%a7%af%e5%88%86%e6%8a%80%e5%b7%a7","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e4%b8%8d%e5%ae%9a%e7%a7%af%e5%88%86%e4%b8%8e%e5%9f%ba%e6%9c%ac%e7%a7%af%e5%88%86%e6%8a%80%e5%b7%a7\/","title":{"rendered":"\u4e0d\u5b9a\u79ef\u5206\u4e0e\u57fa\u672c\u79ef\u5206\u6280\u5de7"},"content":{"rendered":"<style>\n    p, ul, ol {\n        text-align: justify;\n    }\n    h1, h2, h3 {\n    text-align:center;\n    }\n<\/style>\n<p><center><\/p>\n<h1>\u4e0d\u5b9a\u79ef\u5206\u4e0e\u57fa\u672c\u79ef\u5206\u6280\u5de7<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\">\u672c\u8bfe\u4ecb\u7ecd\u4e86\u8ba1\u7b97\u6700\u57fa\u672c\u4e0d\u5b9a\u79ef\u5206\u7684\u57fa\u672c\u6280\u5de7\uff0c\u4ee5\u53ca\u79ef\u5206\u7b97\u5b50\u7684\u6027\u8d28\u3002\u8fd9\u5305\u62ec\u591a\u9879\u5f0f\u79ef\u5206\u3001\u6307\u6570\u51fd\u6570\u3001\u53cc\u66f2\u51fd\u6570\u548c\u57fa\u672c\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5206\u3002<\/p>\n<p style=\"text-align:center;\"><strong><u>\u5b66\u4e60\u76ee\u6807<\/u>\uff1a<\/strong><br \/>\u5728\u672c\u8bfe\u7ed3\u675f\u65f6\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u4e0d\u5b9a\u79ef\u5206\u8fc7\u7a0b\u662f\u6c42\u5bfc\u7684\u9006\u8fc7\u7a0b\u3002<\/li>\n<li><strong>\u8ba1\u7b97<\/strong>\u591a\u9879\u5f0f\u53ca\u6d89\u53ca\u6307\u6570\u51fd\u6570\u3001\u53cc\u66f2\u51fd\u6570\u548c\u4e09\u89d2\u51fd\u6570\u7684\u8868\u8fbe\u5f0f\u7684\u79ef\u5206\u3002<\/li>\n<li><strong>\u8fd0\u7528<\/strong>\u79ef\u5206\u7684\u6027\u8d28\u8fdb\u884c\u4ee3\u6570\u53d8\u6362\uff0c\u4ee5\u7b80\u5316\u79ef\u5206\u8ba1\u7b97\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u5185\u5bb9\u76ee\u5f55<\/strong><br \/>\n<a href=\"#1\">\u4e0d\u5b9a\u79ef\u5206\u7684\u91cd\u8981\u6027<\/a><br \/>\n<a href=\"#2\">\u53cd\u5bfc\u6570\u3001\u4e0d\u5b9a\u79ef\u5206\u4e0e\u51fd\u6570\u7684\u539f\u51fd\u6570<\/a><br \/>\n<a href=\"#3\">\u57fa\u672c\u79ef\u5206\u6280\u5de7<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/4wSTxA7zY9k\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>\u4e0d\u5b9a\u79ef\u5206\u7684\u91cd\u8981\u6027<\/h2>\n<p>\u4e0d\u5b9a\u79ef\u5206\u662f\u5fae\u79ef\u5206\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u5de5\u5177\uff0c\u5728\u7269\u7406\u548c\u6570\u5b66\u79d1\u5b66\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002\u5b83\u4eec\u5141\u8bb8\u6211\u4eec\u8ba1\u7b97\u7ed9\u5b9a\u51fd\u6570\u7684\u539f\u51fd\u6570\uff0c\u800c\u8fd9\u53c8\u53ef\u7528\u4e8e\u8ba1\u7b97\u66f2\u7ebf\u4e0b\u7684\u9762\u79ef\u3001\u56fa\u4f53\u4f53\u79ef\u3001\u6982\u7387\u8ba1\u7b97\u4ee5\u53ca\u5728\u7269\u7406\u3001\u5de5\u7a0b\u3001\u7edf\u8ba1\u548c\u7ecf\u6d4e\u5b66\u4e2d\u7684\u8bb8\u591a\u5176\u4ed6\u5e94\u7528\u3002\u6b64\u5916\uff0c\u4e0d\u5b9a\u79ef\u5206\u5728\u6c42\u89e3\u5fae\u5206\u65b9\u7a0b\u4e2d\u4e5f\u662f\u4e0d\u53ef\u6216\u7f3a\u7684\uff0c\u56e0\u6b64\u5728\u8bb8\u591a\u79d1\u5b66\u4e0e\u6280\u672f\u9886\u57df\u4e2d\u90fd\u81f3\u5173\u91cd\u8981\u3002<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/56fMLiVPwDI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"2\"><\/a><\/p>\n<h2>\u53cd\u5bfc\u51fd\u6570\u3001\u4e0d\u5b9a\u79ef\u5206\u4e0e\u51fd\u6570\u7684\u539f\u51fd\u6570<\/h2>\n<p>\u5982\u679c\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> \u7684\u5bfc\u6570\u662f <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span>\uff0c\u5728\u67d0\u4e2a\u7ed9\u5b9a\u533a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u4e0a\uff0c\u90a3\u4e48\u79f0 <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u5728\u8be5\u533a\u95f4\u4e0a\u7684\u4e00\u4e2a\u539f\u51fd\u6570\u3002<\/p>\n<p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5982\u679c <span class=\"katex-eq\" data-katex-display=\"false\">F(x)<\/span> \u662f <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u7684\u4e00\u4e2a\u539f\u51fd\u6570\uff0c\u90a3\u4e48 <span class=\"katex-eq\" data-katex-display=\"false\">F(x) + C<\/span> \u4e5f\u662f\uff0c\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> \u662f\u4efb\u610f\u5b9e\u5e38\u6570\u3002\u8fd9\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int f(x) dx = F(x) + C<\/span>\n<p>\u5e38\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> \u88ab\u79f0\u4e3a<strong>\u79ef\u5206\u5e38\u6570<\/strong>\uff0c\u5b83\u7684\u5b58\u5728\u8bf4\u660e\u4e00\u4e2a\u51fd\u6570\u7684\u539f\u51fd\u6570\u4e0d\u662f\u4e00\u4e2a\u552f\u4e00\u7684\u51fd\u6570\uff0c\u800c\u662f\u4e00\u65cf\u51fd\u6570\uff1a\u6240\u6709\u5bfc\u6570\u7b49\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> \u7684\u51fd\u6570\u96c6\u5408\uff0c\u5728\u533a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u4e0a\u3002<\/p>\n<p>\u672f\u8bed\u201c\u53cd\u5bfc\u51fd\u6570\u201d\u3001\u201c\u539f\u51fd\u6570\u201d\u548c\u201c\u4e0d\u5b9a\u79ef\u5206\u201d\u8868\u793a\u7684\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4e92\u6362\u4f7f\u7528\u3002\u7b80\u800c\u8a00\u4e4b\uff0c\u4e0d\u5b9a\u79ef\u5206\u662f\u6c42\u5bfc\u7684\u9006\u8fc7\u7a0b\uff0c\u4ece\u8fd9\u4e00\u601d\u60f3\u51fa\u53d1\u53ef\u4ee5\u63a8\u5bfc\u51fa\u5176\u6700\u57fa\u672c\u7684\u6027\u8d28\u3002<\/p>\n<h3>\u4e0d\u5b9a\u79ef\u5206\u7684\u57fa\u672c\u6027\u8d28<\/h3>\n<p>\u4e3a\u4e86\u8ba1\u7b97\u4e0d\u5b9a\u79ef\u5206\uff0c\u6211\u4eec\u9996\u5148\u9700\u8981\u4e86\u89e3\u4e00\u4e9b\u57fa\u672c\u6027\u8d28\uff0c\u8fd9\u4e9b\u6027\u8d28\u76f4\u63a5\u7ee7\u627f\u81ea\u5bfc\u6570\u7684\u6027\u8d28\u3002<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int  \\dfrac{df(x)}{dx} dx = f(x) + C<\/span><\/br>\u56e0\u4e3a\u4e0d\u5b9a\u79ef\u5206\u662f\u6c42\u5bfc\u7684\u9006\u8fc7\u7a0b\u3002<\/li>\n<p><\/br><\/p>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\lambda f(x) dx = \\lambda \\int f(x) dx<\/span><\/br>\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda<\/span> \u662f\u4efb\u610f\u5b9e\u5e38\u6570\u3002\u8fd9\u662f\u56e0\u4e3a\uff1a<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int \\lambda \\dfrac{d\\phi(x)}{dx}dx &amp;=  \\displaystyle \\int \\dfrac{d}{dx}\\lambda \\phi(x) dx \\\\ \\\\\n\n&amp;= \\lambda \\phi(x) + C_1 \\\\ \\\\\n\n&amp;= \\lambda(\\phi(x) + C_2) \\\\ \\\\\n\n&amp;= \\lambda \\displaystyle  \\int \\frac{d\\phi(x)}{dx}dx \\end{array}<\/span><\/center><br \/>\n<\/br><br \/>\n\u7136\u540e\uff0c\u4f7f\u7528 <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{d\\phi(x)}{dx}<\/span> \u5f97\uff1a<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\lambda f(x) dx = \\lambda \\int f(x)dx<\/span><\/center><\/li>\n<p><\/br><\/p>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int f(x) + g(x) dx = \\int f(x) dx + \\int g(x) dx <\/span>\n<\/br><br \/>\n\u8fd9\u4e00\u70b9\u53ef\u4ee5\u7c7b\u4f3c\u5730\u8bc1\u660e\u3002\u6211\u4eec\u8003\u8651\u4e24\u4e2a\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">\\phi(x)<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">\\psi(x)<\/span>\uff0c\u4f7f\u5f97\uff1a<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{d\\phi(x)}{dx}<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\dfrac{d\\psi(x)}{dx}<\/span><\/center><br \/>\n<\/br><br \/>\n\u4e8e\u662f\u6709\uff1a<br \/>\n<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int f(x) + g(x) dx\n\n&amp;= \\displaystyle \\int \\dfrac{d\\phi(x)}{dx} +  \\dfrac{d\\psi(x)}{dx} dx \\\\ \\\\\n\n&amp;= \\displaystyle \\int \\dfrac{d}{dx} (\\phi(x)  + \\psi(x)) dx \\\\ \\\\\n\n&amp;= \\phi(x) + \\psi(x) + C \\\\ \\\\\n\n&amp;= (\\phi(x) + C_1) + (\\psi(x) + C_2) \\\\ \\\\\n\n&amp;= \\displaystyle \\int \\dfrac{d\\phi(x)}{dx} dx + \\int \\dfrac{d\\psi(x)}{dx}dx \\\\ \\\\\n\n&amp;= \\displaystyle \\int f(x) dx + \\int g(x) dx\n\n\\end{array}<\/span><\/center>\n<\/li>\n<\/ol>\n<p><a name=\"3\"><\/a><br \/>\n<\/br><\/br><\/p>\n<h2>\u57fa\u672c\u79ef\u5206\u6280\u5de7<\/h2>\n<p>\u5b58\u5728\u4e00\u4e9b\u57fa\u672c\u7684\u79ef\u5206\u6280\u5de7\uff0c\u53ef\u4ee5\u901a\u8fc7\u5bfc\u6570\u7684\u7ed3\u679c\u6765\u8ba1\u7b97\u67d0\u4e9b\u4e0d\u5b9a\u79ef\u5206\u3002\u5229\u7528\u8fd9\u4e9b\u6280\u5de7\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4ee5\u4e0b\u5bf9\u79ef\u5206\u975e\u5e38\u6709\u7528\u7684\u7ed3\u679c\uff1a<\/p>\n<h3>\u591a\u9879\u5f0f\u51fd\u6570\u7684\u79ef\u5206<\/h3>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int 1 dx = x + C<\/span>\n<\/br><br \/>\n\u56e0\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} (x + C)= 1 <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int x^q dx = \\dfrac{x^{q+1}}{q+1}  + C,<\/span> \u524d\u63d0\u662f <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq -1<\/span>\n<\/br><br \/>\n\u56e0\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx} \\left(\\dfrac{x^{q+1}}{q+1}  + C\\right) = x^q.<\/span>\n<\/li>\n<\/ol>\n<p>\u901a\u8fc7\u8fd9\u4e9b\u7ed3\u679c\u53ca\u57fa\u672c\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u6beb\u4e0d\u8d39\u529b\u5730\u8ba1\u7b97\u4efb\u4f55\u591a\u9879\u5f0f\u7684\u79ef\u5206\u3002<\/p>\n<div style=\"background-color:#F3FFF3; padding:20px;\">\n<p><strong>\u4f8b\u5b50\uff1a<\/strong><\/p>\n<ol>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 3x+2 \\right) dx =  \\dfrac{3}{2}x^2 + 2x + C<\/span><\/li>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 5x^2 + 2x + 3 \\right) dx= \\dfrac{5}{3}x^3 + x + 3x  + C<\/span><\/li>\n<li type=\"a\"> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\left( 4x^{12} - 7x^{-1\/3} + 1 \\right) dx  <\/span> <\/li>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} &amp;= \\dfrac{4}{13}x^{13} - \\dfrac{7}{2\/3}x^{2\/3} + x + C \\\\ \\\\\n\n&amp;= \\dfrac{4}{13}x^{13} - \\dfrac{21}{2}x^{2\/3} + x + C\n\n\\end{array}<\/span>\n<\/ol>\n<\/div>\n<h3>\u6307\u6570\u51fd\u6570\u4e0e\u5bf9\u6570\u51fd\u6570\u7684\u79ef\u5206<\/h3>\n<p>\u6839\u636e\u5df2\u77e5\u7684\u6307\u6570\u51fd\u6570\u548c\u5bf9\u6570\u51fd\u6570\u7684\u5bfc\u6570\u7ed3\u679c\uff0c\u6211\u4eec\u6709\u4ee5\u4e0b\u57fa\u672c\u7ed3\u679c\uff1a<\/p>\n<ol>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int e^{x}dx = e^{x} + C<\/span>\n<br \/>\n\u56e0\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left(e^x + C\\right) = e^x<\/span>\n<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{1}{x} dx = ln|x| + C<\/span>\n<\/br><br \/>\n\u56e0\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left(ln|x| + C \\right) = \\dfrac{1}{|x|} sig(x) = \\dfrac{1}{x}<\/span>\n<\/br><br \/>\n\u5176\u4e2d <span class=\"katex-eq\" data-katex-display=\"false\">sig(x)<\/span> \u662f\u7b26\u53f7\u51fd\u6570\uff0c\u5176\u5b9a\u4e49\u5982\u4e0b\uff1a<br \/>\n<\/br><br \/>\n<center><span class=\"katex-eq\" data-katex-display=\"false\">sig(x) = \\left\\{\\begin{array}{} +1 &amp;,&amp;0\\lt x \\\\ -1 &amp;,&amp; x\\lt 0 \\end{array}\\right.<\/span><\/center>\n<\/li>\n<\/ol>\n<span class=\"katex-eq\" data-katex-display=\"false\">1\/x<\/span> \u7684\u79ef\u5206\u7ed3\u679c\u6269\u5c55\u4e86\u6211\u4eec\u5bf9\u79ef\u5206\u51fd\u6570\u7684\u5904\u7406\u80fd\u529b\uff0c\u56e0\u4e3a\u6211\u4eec\u53ef\u4ee5\u5f00\u59cb\u5bf9\u7531\u591a\u9879\u5f0f\u6784\u6210\u7684\u5206\u5f0f\u51fd\u6570\u8fdb\u884c\u79ef\u5206\u3002<\/p>\n<div style=\"background-color:#F3FFF3; padding:20px;\">\n<p><strong>\u4f8b\u5b50\uff1a<\/strong><\/p>\n<ol>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{x^2 + 3x + 2}{5x^2}dx = \\int \\dfrac{1}{5} + \\dfrac{3}{5}\\dfrac{1}{x} + \\dfrac{2}{5}\\dfrac{1}{x^2}dx<\/span>\n<\/br><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\dfrac{x}{5}+\\dfrac{3}{5}ln(x) - \\dfrac{2}{5}\\dfrac{1}{x} + C <\/span><\/li>\n<p><\/br><\/p>\n<li type=\"a\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int \\dfrac{x^2 - 3 x + 2}{(x-2)^2}dx = \\int \\dfrac{(x-2)^2 + (x-2)}{(x-2)^2} dx<\/span><\/li>\n<p><\/br><br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">= \\displaystyle \\int 1 + \\dfrac{1}{x-2} dx\\\\ \\\\\n\n= x + \\displaystyle \\int \\dfrac{1}{x-2}dx = x + ln|x-2| + C<\/span>\n<\/br><br \/>\n\u56e0\u4e3a<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{d}{dx}\\left( ln|x-2| + C\\right) = \\dfrac{1}{|x-2|}sig(x-2) = \\dfrac{1}{x-2}<\/span>\n<\/ol>\n<\/div>\n<h3>\u57fa\u672c\u53cc\u66f2\u51fd\u6570\u7684\u79ef\u5206<\/h3>\n<p>\u57fa\u672c\u7684\u53cc\u66f2\u51fd\u6570\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} sinh(x) &amp;=&amp; \\dfrac{e^x - e^{-x}}{2} \\\\ \\\\\n\ncosh(x) &amp;=&amp; \\dfrac{e^x + e^{-x}}{2}\n\n\\end{array}<\/span>\n<p>\u7531\u4e8e\u6211\u4eec\u5df2\u7ecf\u4e86\u89e3\u4e86\u6307\u6570\u51fd\u6570\u7684\u79ef\u5206\uff0c\u56e0\u6b64\u5bf9\u53cc\u66f2\u6b63\u5f26\u4e0e\u53cc\u66f2\u4f59\u5f26\u51fd\u6570\u7684\u79ef\u5206\u5c06\u975e\u5e38\u7b80\u5355\u3002<\/p>\n<p>\u5bf9\u4e8e\u53cc\u66f2\u6b63\u5f26\u51fd\u6570\uff0c\u5176\u79ef\u5206\u51e0\u4e4e\u662f\u76f4\u63a5\u7684\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int sinh(x) dx\n\n&amp;=&amp; \\displaystyle \\int \\dfrac{e^x - e^{-x}}{2}dx \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left( \\displaystyle \\int e^x dx - \\int e^{-x}  dx \\right) \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left(e^x + e^{-x} \\right) + C = cosh(x) + C\n\n\\end{array}<\/span>\n<p>\u800c\u5bf9\u4e8e\u53cc\u66f2\u4f59\u5f26\u51fd\u6570\uff0c\u8ba1\u7b97\u51e0\u4e4e\u5b8c\u5168\u7c7b\u4f3c\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int cosh(x) dx\n\n&amp;=&amp; \\displaystyle \\int \\dfrac{e^x + e^{-x}}{2}dx \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left( \\displaystyle \\int e^x dx + \\int e^{-x}  dx \\right) \\\\ \\\\\n\n&amp;=&amp; \\dfrac{1}{2} \\left(e^x - e^{-x} \\right) + C = sinh(x) + C\n\n\\end{array}<\/span>\n<p>\u9664\u6b64\u4e4b\u5916\uff0c\u8fd8\u6709\u5f88\u591a\u5176\u4ed6\u53cc\u66f2\u51fd\u6570\u4e5f\u53ef\u4ee5\u8fdb\u884c\u79ef\u5206\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} tanh(x) &amp;=&amp; \\dfrac{sinh(x)}{cosh(x)} \\\\\n\nsech(x) &amp;=&amp; \\dfrac{1}{cosh(x)} \\\\\n\n{}csch(x) &amp;=&amp; \\dfrac{1}{sinh(x)} \\\\\n\nctgh(x) &amp;=&amp; \\dfrac{1}{tanh(x)}\n\n\\end{array}<\/span>\n<p>\u4e0d\u8fc7\uff0c\u8fd9\u4e9b\u51fd\u6570\u7684\u79ef\u5206\u9700\u8981\u4f7f\u7528\u7a0d\u540e\u5c06\u5b66\u4e60\u7684\u5176\u4ed6\u6280\u5de7\u3002<\/p>\n<h3>\u57fa\u672c\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5206<\/h3>\n<p>\u57fa\u672c\u7684\u4e09\u89d2\u51fd\u6570\u662f <span class=\"katex-eq\" data-katex-display=\"false\">sin(x)<\/span> \u548c <span class=\"katex-eq\" data-katex-display=\"false\">cos(x)<\/span>\u3002\u6839\u636e\u6211\u4eec\u5bf9\u5bfc\u6570\u7684\u8ba4\u8bc6\uff0c\u5b83\u4eec\u7684\u79ef\u5206\u975e\u5e38\u76f4\u63a5\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{} \\displaystyle \\int sin(x) dx = -cos(x) + C \\\\ \\\\\n\n{} \\displaystyle \\int cos(x) dx = sen(x) + C\n\n\\end{array}<\/span>\n<p>\u8fd9\u662f\u56e0\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}\n\n{}  \\dfrac{d}{dx}\\left( sin(x) + C \\right) &amp;=&amp; cos(x) \\\\ \\\\\n\n{}  \\dfrac{d}{dx}\\left( cos(x) + C \\right) &amp;=&amp; -sin(x) \\\\ \\\\\n\n\\end{array}<\/span>\n<h2>\u7ed3\u8bba<\/h2>\n<p>\u672c\u8bfe\u6211\u4eec\u4ece\u7406\u8bba\u57fa\u7840\u5230\u6700\u57fa\u672c\u7684\u5b9e\u9645\u5e94\u7528\uff0c\u5168\u9762\u5730\u63a2\u8ba8\u4e86\u4e0d\u5b9a\u79ef\u5206\u3002\u6211\u4eec\u5b66\u4f1a\u4e86\u5c06\u5176\u89c6\u4e3a\u6c42\u5bfc\u7684\u9006\u8fc7\u7a0b\uff0c\u638c\u63e1\u4e86\u5176\u57fa\u672c\u6027\u8d28\uff0c\u5e76\u80fd\u591f\u76f4\u63a5\u5e94\u7528\u6280\u5de7\u8ba1\u7b97\u591a\u9879\u5f0f\u51fd\u6570\u3001\u6307\u6570\u51fd\u6570\u3001\u5bf9\u6570\u51fd\u6570\u3001\u53cc\u66f2\u51fd\u6570\u4e0e\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5206\u3002\u8fd9\u4e9b\u77e5\u8bc6\u6784\u6210\u4e86\u89e3\u51b3\u66f4\u590d\u6742\u79ef\u5206\u95ee\u9898\u7684\u57fa\u7840\uff0c\u5bf9\u4eca\u540e\u5728\u7269\u7406\u3001\u5de5\u7a0b\u4ee5\u53ca\u5176\u4ed6\u79d1\u5b66\u9886\u57df\u4e2d\u7684\u9ad8\u7ea7\u5e94\u7528\u81f3\u5173\u91cd\u8981\u3002\u638c\u63e1\u8fd9\u4e9b\u57fa\u7840\u77e5\u8bc6\u540e\uff0c\u6211\u4eec\u5c06\u5728\u540e\u7eed\u8bfe\u7a0b\u4e2d\u4ecb\u7ecd\u66f4\u590d\u6742\u7684\u79ef\u5206\u6280\u5de7\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e0d\u5b9a\u79ef\u5206\u4e0e\u57fa\u672c\u79ef\u5206\u6280\u5de7 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