{"id":9177,"date":"2026-04-22T23:28:18","date_gmt":"2026-04-23T06:28:18","guid":{"rendered":"https:\/\/www.assignmenthelp.net\/blog\/?p=9177"},"modified":"2026-04-22T23:28:18","modified_gmt":"2026-04-23T06:28:18","slug":"write-lorenz-equation","status":"publish","type":"post","link":"https:\/\/www.assignmenthelp.net\/blog\/write-lorenz-equation\/","title":{"rendered":"Write lorenz equation"},"content":{"rendered":"\n<p><\/p>\n\n\n<!doctype html>\n<html lang=\"en\">\n<head>\n<meta charset=\"utf-8\"\/>\n<meta content=\"width=device-width, initial-scale=1\" name=\"viewport\"\/>\n<style>         \n          table {\n              border: 1px solid #ccc; \n              border-collapse: collapse; \n              width: 100%; \n              margin: 20px 0; \n              background-color: #f9f9f9;\n          }\n          th, tr {\n            border-bottom: 1px solid #ccc; \n            background-color: #fff;\n          }\n          th, td {\n              border: 1px solid #ccc; \n              color: #333; \n              padding: 12px;\n          }       \n        <\/style>\n<\/head>\n<body>\n<p>Let&#8217;s compute the definite integral:<\/p>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\int_{0}^{\\pi} \\sin^4 x \\, dx<\/annotation><\/semantics><\/math><\/span>\n<hr \/>\n<h3>Step 1: Use a power-reduction identity<\/h3>\n<p>We know that<\/p>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^2 x = \\frac{1 &#8211; \\cos(2x)}{2}<\/annotation><\/semantics><\/math><\/span>\n<p>So,<\/p>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mi>x<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^4 x = (\\sin^2 x)^2 = \\left(\\frac{1 &#8211; \\cos(2x)}{2}\\right)^2<\/annotation><\/semantics><\/math><\/span>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^4 x = \\frac{1}{4}(1 &#8211; 2\\cos(2x) + \\cos^2(2x))<\/annotation><\/semantics><\/math><\/span>\n<hr \/>\n<h3>Step 2: Simplify <span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^2(2x)<\/annotation><\/semantics><\/math><\/span><\/h3>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^2(2x) = \\frac{1 + \\cos(4x)}{2}<\/annotation><\/semantics><\/math><\/span>\n<p>So,<\/p>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><mrow><mo fence=\"true\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^4 x = \\frac{1}{4}\\left(1 &#8211; 2\\cos(2x) + \\frac{1 + \\cos(4x)}{2}\\right)<\/annotation><\/semantics><\/math><\/span>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>3<\/mn><mn>2<\/mn><\/mfrac><mo>\u2212<\/mo><mn>2<\/mn><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"true\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^4 x = \\frac{1}{4}\\left(\\frac{3}{2} &#8211; 2\\cos(2x) + \\frac{1}{2}\\cos(4x)\\right)<\/annotation><\/semantics><\/math><\/span>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>8<\/mn><\/mfrac><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mfrac><mn>1<\/mn><mn>8<\/mn><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^4 x = \\frac{3}{8} &#8211; \\frac{1}{2}\\cos(2x) + \\frac{1}{8}\\cos(4x)<\/annotation><\/semantics><\/math><\/span>\n<hr \/>\n<h3>Step 3: Integrate term by term<\/h3>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><\/mrow><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>3<\/mn><mn>8<\/mn><\/mfrac><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mfrac><mn>1<\/mn><mn>8<\/mn><\/mfrac><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"true\">)<\/mo><\/mrow><mi>d<\/mi><mi>x<\/mi><annotation encoding=\"application\/x-tex\">\\int_{0}^{\\pi} \\sin^4 x \\, dx = \\int_{0}^{\\pi} \\left(\\frac{3}{8} &#8211; \\frac{1}{2}\\cos(2x) + \\frac{1}{8}\\cos(4x)\\right) dx<\/annotation><\/semantics><\/math><\/span>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mo>=<\/mo><mfrac><mn>3<\/mn><mn>8<\/mn><\/mfrac><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><mi>d<\/mi><mi>x<\/mi><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>x<\/mi><mo>+<\/mo><mfrac><mn>1<\/mn><mn>8<\/mn><\/mfrac><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">= \\frac{3}{8}\\int_{0}^{\\pi} dx &#8211; \\frac{1}{2}\\int_{0}^{\\pi}\\cos(2x) dx + \\frac{1}{8}\\int_{0}^{\\pi}\\cos(4x) dx<\/annotation><\/semantics><\/math><\/span>\n<hr \/>\n<h3>Step 4: Evaluate each integral<\/h3>\n<ol>\n<li><span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mi>\u03c0<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\int_{0}^{\\pi} dx = \\pi<\/annotation><\/semantics><\/math><\/span><\/li>\n<li><span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><msubsup><mo fence=\"false\" stretchy=\"true\" minsize=\"1.8em\" maxsize=\"1.8em\">\u2223<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\int_{0}^{\\pi} \\cos(2x) dx = \\frac{\\sin(2x)}{2}\\Big|_{0}^{\\pi} = 0<\/annotation><\/semantics><\/math><\/span><\/li>\n<li><span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><mi>cos<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mi>sin<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mn>4<\/mn><\/mfrac><msubsup><mo fence=\"false\" stretchy=\"true\" minsize=\"1.8em\" maxsize=\"1.8em\">\u2223<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\int_{0}^{\\pi} \\cos(4x) dx = \\frac{\\sin(4x)}{4}\\Big|_{0}^{\\pi} = 0<\/annotation><\/semantics><\/math><\/span><\/li>\n<\/ol>\n<hr \/>\n<h3>Step 5: Combine results<\/h3>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>8<\/mn><\/mfrac><mi>\u03c0<\/mi><mo>\u2212<\/mo><mn>0<\/mn><mo>+<\/mo><mn>0<\/mn><mo>=<\/mo><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><\/mstyle><mstyle scriptlevel=\"0\" displaystyle=\"false\"><\/mstyle><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mfrac><mrow><mn>3<\/mn><mi>\u03c0<\/mi><\/mrow><mn>8<\/mn><\/mfrac><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\int_{0}^{\\pi} \\sin^4 x \\, dx = \\frac{3}{8}\\pi &#8211; 0 + 0 = \\boxed{\\frac{3\\pi}{8}}<\/annotation><\/semantics><\/math><\/span>\n<hr \/>\n<p>\u2705 <strong>Final Answer:<\/strong><\/p>\n<span class=\"katex\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><msubsup><mo>\u222b<\/mo><mn>0<\/mn><mi>\u03c0<\/mi><\/msubsup><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mn>4<\/mn><\/msup><mi>x<\/mi><mtext>\u2009<\/mtext><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mn>3<\/mn><mi>\u03c0<\/mi><\/mrow><mn>8<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\int_{0}^{\\pi} \\sin^4 x \\, dx = \\frac{3\\pi}{8}<\/annotation><\/semantics><\/math><\/span>\n<hr \/>\n<p>Would you like me to <strong>put this full solution<\/strong> into your Word template and generate a formatted <code>.docx<\/code> file for download?<\/p>\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Let&#8217;s compute the definite integral: \u222b0\u03c0sin\u20614x\u2009dx\\int_{0}^{\\pi} \\sin^4 x \\, dx Step 1: Use a power-reduction identity We know that sin\u20612x=1\u2212cos\u2061(2x)2\\sin^2 x = \\frac{1 &#8211; \\cos(2x)}{2} So, sin\u20614x=(sin\u20612x)2=(1\u2212cos\u2061(2x)2)2\\sin^4 x = (\\sin^2 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[26],"tags":[],"class_list":["post-9177","post","type-post","status-publish","format-standard","hentry","category-assignment-help","has_no_thumb"],"_links":{"self":[{"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/posts\/9177","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/comments?post=9177"}],"version-history":[{"count":1,"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/posts\/9177\/revisions"}],"predecessor-version":[{"id":9178,"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/posts\/9177\/revisions\/9178"}],"wp:attachment":[{"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/media?parent=9177"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/categories?post=9177"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.assignmenthelp.net\/blog\/wp-json\/wp\/v2\/tags?post=9177"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}