{"id":11227,"date":"2023-08-29T17:02:59","date_gmt":"2023-08-29T09:02:59","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=11227"},"modified":"2025-07-27T01:21:20","modified_gmt":"2025-07-26T17:21:20","slug":"%e7%a7%af%e5%88%86","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=11227","title":{"rendered":"\u79ef\u5206"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: 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class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E5%85%AC%E5%BC%8F\" >\u516c\u5f0f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E6%96%B9%E6%B3%95\" >\u65b9\u6cd5<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E5%87%91%E5%BE%AE%E5%88%86%E6%B3%95%EF%BC%88%E7%AC%AC%E4%B8%80%E6%8D%A2%E5%85%83%E6%B3%95%EF%BC%89\" >\u51d1\u5fae\u5206\u6cd5\uff08\u7b2c\u4e00\u6362\u5143\u6cd5\uff09<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E6%8D%A2%E5%85%83%E7%A7%AF%E5%88%86%E6%B3%95%EF%BC%88%E7%AC%AC%E4%BA%8C%E6%8D%A2%E5%85%83%E6%B3%95%EF%BC%89\" >\u6362\u5143\u79ef\u5206\u6cd5\uff08\u7b2c\u4e8c\u6362\u5143\u6cd5\uff09<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E5%B8%B8%E8%A7%81%E5%87%A0%E7%A7%8D%E5%85%B8%E5%9E%8B%E7%B1%BB%E5%9E%8B%E7%9A%84%E6%8D%A2%E5%85%83%E6%96%B9%E6%B3%95\" >\u5e38\u89c1\u51e0\u79cd\u5178\u578b\u7c7b\u578b\u7684\u6362\u5143\u65b9\u6cd5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95\" >\u5206\u90e8\u79ef\u5206\u6cd5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#Wallis_%E5%85%AC%E5%BC%8F\" >Wallis \u516c\u5f0f<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E9%83%A8%E5%88%86%E5%90%AB%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E7%A7%AF%E5%88%86%E7%9A%84%E4%BB%A3%E6%8D%A2%E5%85%B3%E7%B3%BB\" >\u90e8\u5206\u542b\u4e09\u89d2\u51fd\u6570\u79ef\u5206\u7684\u4ee3\u6362\u5173\u7cfb<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E5%91%A8%E6%9C%9F%E5%87%BD%E6%95%B0\" >\u5468\u671f\u51fd\u6570<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E5%81%B6%E5%87%BD%E6%95%B0\" >\u5076\u51fd\u6570<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/blog.iyatt.com\/?p=11227\/#%E5%B8%B8%E7%94%A8%E6%9B%B2%E7%BA%BF\" >\u5e38\u7528\u66f2\u7ebf<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"%E6%80%A7%E8%B4%A8\"><\/span>\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86%E7%9A%84%E6%80%A7%E8%B4%A8\"><\/span>\u4e0d\u5b9a\u79ef\u5206\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\u2460(\\int f(x)dx)&#039;=f(x) \\\\\n\u2461\\int f&#039;(x)dx=f(x)+C \\\\\n\u2462\\int(f(x)\\pm g(x))=\\int f(x)dx\\pm\\int g(x)dx \\\\\n\u2463\\int kf(x)dx=k\\int f(x)dx\uff0c\u5e38\u6570 k\\ne0\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E5%AE%9A%E7%A7%AF%E5%88%86%E7%9A%84%E6%80%A7%E8%B4%A8\"><\/span>\u5b9a\u79ef\u5206\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\u2460\\int_a^bf(x)dx=-\\int_b^af(x)dx \\\\\n\u2461\\int_a^af(x)dx=0 \\\\\n\u2462\\int_a^b[f(x)\\pm g(x)]dx=\\int_a^bf(x)dx\\pm\\int_a^bg(x)dx \\\\\n\u2463\\int_a^b kf(x)dx=k\\int_a^bf(x)dx\uff0ck\u4e3a\u5e38\u6570 \\\\\n\u2464\\int_a^bf(x)dx=\\int_a^cf(x)dx+\\int_c^bf(x)dx \\\\\n\u2465\u5b9a\u79ef\u5206\u7684\u4e0d\u7b49\u5f0f\u6027\u8d28 \\\\\n\u82e5f(x)\\le g(x)\uff0ca\\le b\uff0c\u5219\\int_a^bf(x)dx\\le\\int_a^bg(x)dx \\\\\n\u82e5f(x)\u4e0eg(x)\u5728[a,b]\u8fde\u7eed\uff0cf(x)\\le g(x)\uff0c\u4e14\u81f3\u5c11\u5b58\u5728\u4e00\u70b9x_1\uff0ca\\le x_1\\le b\uff0c\u4f7f\u5f97f(x_1)\\lt g(x_1)\uff0c\u90a3\u4e48\u6709\\int_a^bf(x)\\lt\\int_a^bg(x)\u3002\\\\\n\u2466\u79ef\u5206\u4e2d\u503c\u5b9a\u7406 \\\\\n\u8bbef(x)\u5728[a,b]\u8fde\u7eed\uff0c\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\\xi\\in(a,b)\u4f7f\\int_a^bf(x)dx=f(\\xi)(b-a)\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E5%AE%9A%E7%A7%AF%E5%88%86%E5%AD%98%E5%9C%A8%E5%AE%9A%E7%90%86\"><\/span>\u5b9a\u79ef\u5206\u5b58\u5728\u5b9a\u7406<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\u8bbef(x)\u5728[a,b]\u8fde\u7eed\uff0c\u5219\\int_a^bf(x)dx\u5b58\u5728\u3002 \\\\\n\u8bbef(x)\u5728[a,b]\u6709\u754c\uff0c\u4e14\u53ea\u6709\u6709\u9650\u4e2a\u95f4\u65ad\u70b9\uff0c\u5219\\int_a^bf(x)dx\u5b58\u5728\u3002\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E5%8E%9F%E5%87%BD%E6%95%B0%E5%AD%98%E5%9C%A8%E5%AE%9A%E7%90%86\"><\/span>\u539f\u51fd\u6570\u5b58\u5728\u5b9a\u7406<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8bbef(x)\u5728[a,b]\u4e0a\u8fde\u7eed\uff0c\u5219\u5728[a,b]\u4e0a\u5fc5\u5b58\u5728\u539f\u51fd\u6570\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E5%8F%98%E9%99%90%E7%A7%AF%E5%88%86\"><\/span>\u53d8\u9650\u79ef\u5206<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8bbef(x)\u5728[a,b]\u4e0a\u53ef\u79ef\uff0c\u5bf9<code class=\"katex-inline\">x\\in[a,b]<\/code>\uff0cf(x)\u5728[a,x]\u4e0a\u53ef\u79ef\u3002<br \/>\n<code class=\"katex-inline\">\\Phi=\\int_a^xf(x)dx\uff0cx\\in[a,b]<\/code><br \/>\n\u5b9a\u4e49\u4e86\u4e00\u4e2a\u4ee5x\u4e3a\u81ea\u53d8\u91cf\u7684\u51fd\u6570\uff0c\u79f0\u4e3a<strong>\u53d8\u4e0a\u9650\u7684\u5b9a\u79ef\u5206<\/strong>\u3002<br \/>\n\u5982\u679c\u4e0b\u9650\u662f\u81ea\u53d8\u91cf\uff0c\u5219\u79f0\u4e3a<strong>\u53d8\u4e0b\u9650\u7684\u5b9a\u79ef\u5206<\/strong>\uff0c\u4e24\u7c7b\u7edf\u79f0\u4e3a\u53d8\u9650\u79ef\u5206\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E5%8F%98%E9%99%90%E5%AE%9A%E7%A7%AF%E5%88%86%E6%B1%82%E5%AF%BC\"><\/span>\u53d8\u9650\u5b9a\u79ef\u5206\u6c42\u5bfc<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8bbef(x)\u5728[a,b]\u4e0a\u8fde\u7eed\uff0c<code class=\"katex-inline\">\\int_a^bf(t)dt<\/code>\u662ff(x)\u7684\u4e00\u4e2a\u539f\u51fd\u6570\uff0c\u4ece\u800c\u6709<code class=\"katex-inline\">\\int f(x)dx=\\int_a^xf(t)dt+C<\/code>\u3002<\/p>\n<p>\u53d8\u4e0a\u9650\u5b9a\u79ef\u5206\u6c42\u5bfc\uff1a<code class=\"katex-inline\">(\\int_a^{g(x)}f(t)dt)'=f(g(x))\\cdot g'(x)<\/code><\/p>\n<p>\u53d8\u4e0b\u9650\u5b9a\u79ef\u5206\u6c42\u5bfc\uff1a<code class=\"katex-inline\">(\\int_{h(x)}^bf(t)dt)'=-f(h(x))\\cdot h'(x)<\/code><\/p>\n<p>\u53d8\u9650\u5b9a\u79ef\u5206\u6c42\u5bfc\uff1a<code class=\"katex-inline\">(\\int_{h(x)}^{g(x)}f(t)dt)'=f(g(x))\\cdot g'(x)-f(h(x))\\cdot h'(x)<\/code><\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E7%89%9B%E9%A1%BF-%E8%8E%B1%E5%B8%83%E5%B0%BC%E8%8C%A8%E5%AE%9A%E7%90%86\"><\/span>\u725b\u987f-\u83b1\u5e03\u5c3c\u8328\u5b9a\u7406<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8bbef(x)\u5728[a,b]\u8fde\u7eed\uff0cF(x)\u662ff(x)\u7684\u4e00\u4e2a\u539f\u51fd\u6570\uff0c\u5219<code class=\"katex-inline\">\\int_a^bf(x)dx=F(x)|_a^b<\/code>=F(b)-F(a)\u3002<\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E8%AE%A1%E7%AE%97\"><\/span>\u8ba1\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"%E5%85%AC%E5%BC%8F\"><\/span>\u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l l}\n\\int0dx=C &amp; \\int1dx=x+C \\\\\n\\int x^adx=\\frac{1}{a+1}a^{x+1}\uff08a\\ne-1\uff09 &amp; \\int\\frac{1}{x}dx=\\ln|x|+C \\\\\n\\int a^xdx=\\frac{a^x}{\\ln a}+C\uff08a\\gt0\uff0ca\\ne1\uff09&amp; \\int e^x=e^x+C \\\\\n\\int \\sin xdx=-\\cos x+C &amp;  \\int \\cos xdx=\\sin x+C \\\\\n\\int \\tan xdx=-\\ln|\\cos x|+C &amp; \\int \\cot xdx=\\ln|\\sin x|+C \\\\\n\\int \\sec xdx=\\ln(\\sec x+\\tan x)+C &amp; \\int\\csc xdx=\\ln|\\csc x-\\cot x|+C \\\\\n\\int \\sec^2xdx=\\tan x+C &amp; \\int\\csc^2xdx=-\\cot x+C \\\\\n\\int\\frac{1}{a^2+x^2}dx=\\frac{1}{a}\\arctan\\frac{x}{a}+C &amp; \\int\\frac{1}{a^2-x^2}dx=\\frac{1}{2a}\\ln|\\frac{a+x}{a-x}|+C \\\\\n\\int\\frac{1}{\\sqrt{a^2-x^2}}dx=\\arcsin\\frac{x}{a}+C &amp; \\int\\frac{dx}{\\sqrt{x^2\\pm a^2}}=\\ln|x+\\sqrt{x^2\\pm a^2}|+C \\\\\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95\"><\/span>\u65b9\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"%E5%87%91%E5%BE%AE%E5%88%86%E6%B3%95%EF%BC%88%E7%AC%AC%E4%B8%80%E6%8D%A2%E5%85%83%E6%B3%95%EF%BC%89\"><\/span>\u51d1\u5fae\u5206\u6cd5\uff08\u7b2c\u4e00\u6362\u5143\u6cd5\uff09<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u8bbef(u)\u8fde\u7eed\uff0c<code class=\"katex-inline\">\\phi(x)<\/code>\u5177\u6709\u8fde\u7eed\u7684\u4e00\u9636\u5bfc\u6570\uff0c\u5219\u6709<code class=\"katex-inline\">\\int f[\\phi(x)]\\phi'(x)dx=\\int f[\\phi(x)]d\\phi(x)\\xlongequal{\u4ee4\\phi(x)=u}=\\int f(u)du\\xlongequal{\\int f(t)dt=F(t)+C}F(u)+C=F(\\phi(x))+C<\/code><\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E6%8D%A2%E5%85%83%E7%A7%AF%E5%88%86%E6%B3%95%EF%BC%88%E7%AC%AC%E4%BA%8C%E6%8D%A2%E5%85%83%E6%B3%95%EF%BC%89\"><\/span>\u6362\u5143\u79ef\u5206\u6cd5\uff08\u7b2c\u4e8c\u6362\u5143\u6cd5\uff09<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u8bbef(x)\u8fde\u7eed\uff0c<code class=\"katex-inline\">x=\\phi(t)<\/code>\u5177\u6709\u8fde\u7eed\u5bfc\u6570<code class=\"katex-inline\">\\phi'(t)<\/code>\uff0c\u4e14<code class=\"katex-inline\">\\phi'(t)\\ne0<\/code>\uff0c\u5219<code class=\"katex-inline\">\\int f(x)dx\\xlongequal{x=\\phi(t)}(\\int f(\\phi(t))\\phi'(t)dt)_{t=\\psi(x)}<\/code><br \/>\n\u5373\u5bf9t\u79ef\u5206\u540e\uff0c\u518d\u4ee5<code class=\"katex-inline\">x=\\phi(t)<\/code>\u7684\u53cd\u51fd\u6570<code class=\"katex-inline\">t=\\psi(x)<\/code>\u4ee3\u56dex\u7684\u51fd\u6570\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E5%B8%B8%E8%A7%81%E5%87%A0%E7%A7%8D%E5%85%B8%E5%9E%8B%E7%B1%BB%E5%9E%8B%E7%9A%84%E6%8D%A2%E5%85%83%E6%96%B9%E6%B3%95\"><\/span>\u5e38\u89c1\u51e0\u79cd\u5178\u578b\u7c7b\u578b\u7684\u6362\u5143\u65b9\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\u542b\\sqrt{a^2-x^2}\uff0c\u4ee4x=a\\sin t\uff0cdx=a\\cos tdt \\\\\n\u542b\\sqrt{x^2+a^2}\uff0c\u4ee4x=a\\tan t\uff0cdx=a\\sec^2tdt \\\\\n\u542b\\sqrt{x^2-a^2}\uff0c\u4ee4x=a\\sec t\uff0cdx=a\\sec t\\tan tdt \\\\\n\\\\\n\u542b\\sqrt[n]{ax+b}\uff0c\u4ee4\\sqrt[n]{ax+b}=t\uff0cx=\\frac{t^n-b}{a}\uff0cdx=\\frac{n}{a}t^{n-1}dt \\\\\n\\\\\n\u542b\\sqrt{\\frac{ax+b}{cx+d}}\uff0c\u4ee4\\sqrt{\\frac{ax+b}{cx+d}}=t\uff0cx=\\frac{dt^2-b}{a-ct^2}\uff0cdx=\\frac{2(ab-bc)t}{(a-ct^2)^2}dt \\\\\n\\\\\n\u542b\u6b63\u5f26\u4f59\u5f26\u7684\uff0c\u4ee4\\tan\\frac{x}{2}=t\uff0c\u5219\\sin x=\\frac{2t}{1+t^2}\uff0c\\cos x=\\frac{1-t^2}{1+t^2}\uff0cdx=\\frac{2}{1+t^2}dt \\\\\n\u6b64\u4e3a\u4e07\u80fd\u4ee3\u6362\uff0c\u4f46\u662f\u4e00\u822c\u8ba1\u7b97\u4f1a\u8f83\u4e3a\u590d\u6742\uff0c\u5efa\u8bae\u4f5c\u4e3a\u6700\u540e\u624b\u6bb5 \\\\\n\\end{array}<\/code><\/pre>\n<h3><span class=\"ez-toc-section\" id=\"%E5%88%86%E9%83%A8%E7%A7%AF%E5%88%86%E6%B3%95\"><\/span>\u5206\u90e8\u79ef\u5206\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><code class=\"katex-inline\">\\int udv=uv-\\int vdu<\/code><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Wallis_%E5%85%AC%E5%BC%8F\"><\/span>Wallis \u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><a href=\"https:\/\/blog.iyatt.com\/?p=11779\">https:\/\/blog.iyatt.com\/?p=11779<\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E9%83%A8%E5%88%86%E5%90%AB%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E7%A7%AF%E5%88%86%E7%9A%84%E4%BB%A3%E6%8D%A2%E5%85%B3%E7%B3%BB\"><\/span>\u90e8\u5206\u542b\u4e09\u89d2\u51fd\u6570\u79ef\u5206\u7684\u4ee3\u6362\u5173\u7cfb<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\\int_0^\\frac{\\pi}{2}f(\\sin x)dx=\\int_0^\\frac{\\pi}{2}f(\\cos x)dx \\\\\n\\int_0^\\pi f(\\sin x)dx=2\\int_0^\\frac{\\pi}{2}f(\\sin x)dx \\\\\n\\int_0^\\pi xf(\\sin x)dx=\\frac{\\pi}{2}\\int_0^\\pi f(\\sin x)dx\n\\end{array}<\/code><\/pre>\n<p>\u5176\u4e2d <code class=\"katex-inline\">f(\\sin x)<\/code> \u53ef\u4ee5\u5305\u542b <code class=\"katex-inline\">\\sin x\u3001|\\cos x|\u3001\\cos^n x\uff08n \u4e3a\u5076\u6570\uff09<\/code><\/p>\n<p>\u4f8b\uff1a<br \/>\n<code class=\"katex-inline\">I=\\int_0^\\pi\\frac{x|\\sin x\\cos x|}{1+\\cos^2x}dx<\/code><\/p>\n<pre><code class=\"language-katex\">\\begin{aligned}\nI&amp;=\\frac{\\pi}{2}\\int_0^\\pi\\frac{|\\sin x\\cos x|}{1+\\cos^2x}dx \\\\\n&amp;=\\pi\\int_0^\\frac{\\pi}{2}\\frac{\\sin x\\cos x}{1+\\cos^2x}dx \\\\\n&amp;=-\\frac{\\pi}{2}\\int_0^\\frac{\\pi}{2}\\frac{1}{1+\\cos^2x}d(1+\\cos^2x) \\\\\n&amp;=-\\frac{\\pi}{2}\\ln(1+\\cos^2x)|_0^\\frac{\\pi}{2} \\\\\n&amp;=\\frac{\\pi}{2}\\ln2\n\\end{aligned}<\/code><\/pre>\n<h3><span class=\"ez-toc-section\" id=\"%E5%91%A8%E6%9C%9F%E5%87%BD%E6%95%B0\"><\/span>\u5468\u671f\u51fd\u6570<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\\int_a^bf(x)dx=\\int_{a+t}^{b+t}f(x)dx\uff0cf(x) \u7684\u4e00\u4e2a\u5468\u671f\u4e3a T=|b-a| \\\\\n\\\\\n\u5982\uff1a\\\\\n\\int_0^{2\\pi}\\sin xdx=\\int_\\pi^{3\\pi}\\sin xdx\n\\end{array}<\/code><\/pre>\n<h3><span class=\"ez-toc-section\" id=\"%E5%81%B6%E5%87%BD%E6%95%B0\"><\/span>\u5076\u51fd\u6570<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">f(x) \u4e3a\u5076\u51fd\u6570\uff0c\\int_{-a}^af(x)dx=2\\int_0^af(x)dx=2\\int_{-a}^0f(x)dx<\/code><\/pre>\n<h3><span class=\"ez-toc-section\" id=\"%E5%B8%B8%E7%94%A8%E6%9B%B2%E7%BA%BF\"><\/span>\u5e38\u7528\u66f2\u7ebf<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><a href=\"https:\/\/blog.iyatt.com\/?p=11803\">https:\/\/blog.iyatt.com\/?p=11803<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6027\u8d28 \u4e0d\u5b9a\u79ef\u5206\u7684\u6027\u8d28 \\begin{array}{l} \u2460(\\int f(x)dx)&#039;=f(x) \\ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,612],"tags":[],"class_list":["post-11227","post","type-post","status-publish","format-standard","hentry","category-all","category-612"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11227","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11227"}],"version-history":[{"count":8,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11227\/revisions"}],"predecessor-version":[{"id":21073,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/11227\/revisions\/21073"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}