{"id":6807,"date":"2021-10-21T17:49:02","date_gmt":"2021-10-21T12:19:02","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6807"},"modified":"2021-11-01T19:31:36","modified_gmt":"2021-11-01T14:01:36","slug":"evaluate-int-dxover-3sinx-4cosx","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/","title":{"rendered":"Evaluate : \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\)"},"content":{"rendered":"

Solution :<\/h2>\n

I = \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\) = \\(\\int\\) \\(dx\\over {3[{2tan{x\\over 2}\\over {1+tan^2{x\\over 2}}}] + 4[{1-tan^2{x\\over 2}\\over {1+tan^2{x\\over 2}}}]}\\) = \\(\\int\\) \\(sec^2{x\\over 2}dx\\over {4+6tan{x\\over 2}-4tan^2{x\\over 2}}\\)<\/p>\n

let \\(tan{x\\over 2}\\) = t,<\/p>\n

\\(\\therefore\\)\u00a0 \\({1\\over 2}sec^2{x\\over 2}\\)dx = dt<\/p>\n

so I = \\(\\int\\) \\(2dt\\over {4+6t-4t^2}\\) = \\(1\\over 2\\) \\(\\int\\) \\(dt\\over {1-(t^2-{3\\over 2}t})\\) = \\(1\\over 2\\) \\(\\int\\) \\(dt\\over {{25\\over 16}-{(t-{3\\over 4})}^2}\\)<\/p>\n

= \\(1\\over 2\\) \\(1\\over {2({5\\over 4})}\\) \\(ln|{{{5\\over 4}+(t-{3\\over 4})}\\over {{5\\over 4}-(t-{3\\over 4})}}|\\) + C = \\(1\\over 5\\) \\(ln|{1+2tan{x\\over 2}\\over {4-2tan{x\\over 2}}}|\\) + C<\/p>\n

\n

Similar Questions<\/h3>\n

What is the integration of x tan inverse x dx ?<\/a><\/p>\n

Prove that \\(\\int_{0}^{\\pi\/2}\\) log(sinx)dx = \\(\\int_{0}^{\\pi\/2}\\) log(cosx)dx = -\\(\\pi\\over 2\\)log 2.<\/a><\/p>\n

Evaluate : \\(\\int\\) \\(cos^4xdx\\over {sin^3x{(sin^5x + cos^5x)^{3\\over 5}}}\\)<\/a><\/p>\n

What is the integration of tan inverse root x ?<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : I = \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\) = \\(\\int\\) \\(dx\\over {3[{2tan{x\\over 2}\\over {1+tan^2{x\\over 2}}}] + 4[{1-tan^2{x\\over 2}\\over {1+tan^2{x\\over 2}}}]}\\) = \\(\\int\\) \\(sec^2{x\\over 2}dx\\over {4+6tan{x\\over 2}-4tan^2{x\\over 2}}\\) let \\(tan{x\\over 2}\\) = t, \\(\\therefore\\)\u00a0 \\({1\\over 2}sec^2{x\\over 2}\\)dx = dt so I = \\(\\int\\) \\(2dt\\over {4+6t-4t^2}\\) = \\(1\\over 2\\) \\(\\int\\) \\(dt\\over {1-(t^2-{3\\over 2}t})\\) = \\(1\\over 2\\) …<\/p>\n

Evaluate : \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\)<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[52,43],"tags":[],"yoast_head":"\nEvaluate : \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\)<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Evaluate : \\(\\int\\) \\(dx\\over {3sinx + 4cosx}\\)\" \/>\n<meta property=\"og:description\" content=\"Solution : I = (int) (dxover {3sinx + 4cosx}) = (int) (dxover {3[{2tan{xover 2}over {1+tan^2{xover 2}}}] + 4[{1-tan^2{xover 2}over {1+tan^2{xover 2}}}]}) = (int) (sec^2{xover 2}dxover {4+6tan{xover 2}-4tan^2{xover 2}}) let (tan{xover 2}) = t, (therefore)\u00a0 ({1over 2}sec^2{xover 2})dx = dt so I = (int) (2dtover {4+6t-4t^2}) = (1over 2) (int) (dtover {1-(t^2-{3over 2}t})) = (1over 2) … Evaluate : (int) (dxover {3sinx + 4cosx}) Read More »\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-21T12:19:02+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-01T14:01:36+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Evaluate : \\\\(\\\\int\\\\) \\\\(dx\\\\over {3sinx + 4cosx}\\\\)\",\"datePublished\":\"2021-10-21T12:19:02+00:00\",\"dateModified\":\"2021-11-01T14:01:36+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/\"},\"wordCount\":152,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"articleSection\":[\"Integration Questions\",\"Maths Questions\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/\",\"url\":\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/\",\"name\":\"Evaluate : \\\\(\\\\int\\\\) \\\\(dx\\\\over {3sinx + 4cosx}\\\\)\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-21T12:19:02+00:00\",\"dateModified\":\"2021-11-01T14:01:36+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/evaluate-int-dxover-3sinx-4cosx\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Evaluate : \\\\(\\\\int\\\\) \\\\(dx\\\\over {3sinx + 4cosx}\\\\)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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