{"id":11672,"date":"2021-02-25T01:24:32","date_gmt":"2021-02-24T19:54:32","guid":{"rendered":"https:\/\/learnsteer.sasnaka.org\/science\/?p=11672"},"modified":"2021-11-12T08:00:16","modified_gmt":"2021-11-12T02:30:16","slug":"04-05-04","status":"publish","type":"post","link":"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/combined-mathematics\/04-05-04\/","title":{"rendered":"04.05.04 &#8211; \u0dc0\u0dd2\u0da0\u0dbd\u0dca\u200d\u0dba \u0db8\u0dcf\u0dbb\u0dd4 \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dd2\u0dbb\u0dd3\u0db8"},"content":{"rendered":"\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button is-style-shadow\"><a class=\"wp-block-button__link\" href=\"https:\/\/drive.google.com\/uc?id=1zaYSXmmKtEI7L634oT3ufxFF4YbXDqbX&amp;export=download\" style=\"border-radius:15px\" target=\"_blank\" rel=\"noreferrer noopener\">\u0db4\u0dcf\u0da9\u0db8\u0dda \u0dc3\u0da7\u0dc4\u0db1 Download \u0d9a\u0dbb\u0d9c\u0db1\u0dca\u0db1.<\/a><\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>\u0dc0\u0dd2\u0da0\u0dbd\u0dca\u200d\u0dba \u0db8\u0dcf\u0dbb\u0dd4 \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dd2\u0dbb\u0dd3\u0db8<\/strong><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u0db4\u0dca\u200d\u0dbb\u0db8\u0dda\u0dba\u0dba <\/strong><\/h3>\n\n\n<h3 style=\"text-align: center\"><strong><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\left\\{f\\left(x\\right)\\right\\}^nf&#039;\\left(x\\right)dx=\\frac{\\left\\{f\\left(x\\right)\\right\\}^{n+1}}{n+1}+c<\/span><\/strong><\/h3>\n\n\n\n\n<p>&nbsp;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u0dc3\u0dcf\u0db0\u0db1\u0dba&nbsp;<\/strong>&nbsp;<\/h3>\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac d{dx}\\left[\\frac{\\left\\{f\\left(x\\right)\\right\\}^{n+1}}{n+1}+c\\right]&amp;=&amp;\\frac1{\\left(n+1\\right)}.(n+1)\\left\\{f\\left(x\\right)\\right\\}^{n+1-1}\\frac d{dx}f\\left(x\\right)+0\\\\&amp;=&amp;\\left\\{f\\left(x\\right)\\right\\}^n.f&#039;\\left(x\\right)\\end{array}<\/span>\n\n\n\n\n\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d85\u0dbb\u0dca\u0dae \u0daf\u0dd0\u0d9a\u0dca\u0dc0\u0dd3\u0db8\u0da7 \u0d85\u0db1\u0dd4\u0dc0,<\/li><\/ul>\n\n\n\n<p>&nbsp;&nbsp;&nbsp;&nbsp; <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\left\\{f\\left(x\\right)\\right\\}^nf&#039;\\left(x\\right)dx=\\frac{\\left\\{f\\left(x\\right)\\right\\}^{n+1}}{n+1}+c<\/span>&nbsp;\u0dc0\u0dda.<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d8b\u0daf\u0dcf\u0dc4\u0dbb\u0dab 01<\/li><\/ul>\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\int\\sin^3x\\cos dx\\\\\\frac d{dx}\\left(\\sin x\\right)=\\cos x\\Rightarrow d\\left(\\sin x\\right)=\\cos xdx\\\\\\int\\sin^3x\\cos dx=\\int\\sin^3xd\\left(\\sin x\\right)\\\\{=\\frac14\\sin^4x+c}\\\\\\\\\\end{array}<\/span>\n\n\n\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d8b\u0daf\u0dcf\u0dc4\u0dbb\u0dab 02<\/li><\/ul>\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\int x^3e^{x^4}dx\\\\\\\\\\end{array}<\/span>\n<p style=\"text-align: left\"><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\frac d{dx}\\left(x^4\\right)=4x^3\\Rightarrow d\\left(x^4\\right)=4x^3dx\\\\\\\\\\end{array}<\/span><\/p>\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int x^3e^{x^4}dx&amp;=&amp;\\frac14\\int e^{x^4}d\\left(x^4\\right)\\\\&amp;=&amp;\\frac14e^{x^4}+c\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>&nbsp; ; c- \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2<\/p>\n\n\n\n\n\n\n<p>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d8b\u0daf\u0dcf\u0dc4\u0dbb\u0dab 03<\/li><\/ul>\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\int e^{\\cos x}\\sin xdx\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n<div class=\"td-paragraph-padding-0\">\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac d{dx}\\left(\\cos x\\right)&amp;=&amp;-\\sin x\\Rightarrow d\\left(\\cos x\\right)=-\\sin xdx\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n<\/div>\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int e^{\\cos x}\\sin xdx&amp;=&amp;-\\int e^{\\cos x}d\\left(\\cos x\\right)\\\\&amp;=&amp;-e^{\\cos x}+c\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n\n\n\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d8b\u0daf\u0dcf\u0dc4\u0dbb\u0dab 04<\/li><\/ul>\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\int e^x(1-e^x)^3dx\\\\\\frac d{dx}(e^x)=e^x\\;\\Rightarrow\\;e^xdx\\\\\\end{array}<\/span>\n<p style=\"text-align: left\"><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\therefore\\int e^x(1-e^x)^3dx&amp;=&amp;\\int(1-e^x)^3.d(e^x)\\\\&amp;=&amp;\\int d(e^x)-3\\int e^xd(e^x)+3\\int e^{x^2}d(e^x)-{\\int e^{x^3}d(e^x})\\\\&amp;=&amp;e^x-\\frac32e^{x^2}+e^{x^3}-\\frac14e^{x^4}+C\\;(C\\;\u0d85\u0db7\u0dd2\u0db8\u0dad\\;\u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2)\\end{array}<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d8b\u0daf\u0dcf\u0dc4\u0dbb\u0dab 05<\/li><\/ul>\n\n\n<p style=\"text-align: left\"><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\int\\left(2x+1\\right)e^{\\left(x^2+x\\right)}dx\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span><\/p>\n<p style=\"text-align: left\"><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\\\\\frac d{dx}\\left(x^2+x\\right)&amp;=&amp;2x+1\\Rightarrow d\\left(x^2+x\\right)=\\left(2x+1\\right)dx\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span><\/p>\n<p style=\"text-align: left\"><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\left(2x+1\\right)e^{\\left(x^2+x\\right)}dx&amp;=&amp;\\int e^{\\left(x^2+x\\right)}d\\left(x^2+x\\right)\\\\&amp;=&amp;\\frac12e^{\\left(x^2+x\\right)}+c\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span><\/p>\n\n\n\n\n\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d8b\u0daf\u0dcf\u0dc4\u0dbb\u0dab 06<\/li><\/ul>\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\int\\left(\\frac{\\sin x+\\cos x}{\\cos x-\\sin x}\\right)\\left(2+2\\sin2x\\right)dx\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac d{dx}\\left(\\cos x-\\sin x\\right)&amp;=&amp;-\\left(\\sin x+\\cos x\\right)\\Rightarrow d\\left(\\cos x-\\sin x\\right)=-\\left(\\sin x+\\cos x\\right)dx\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\left(\\frac{\\sin x+\\cos x}{\\cos x-\\sin x}\\right)\\left(2+2\\sin2x\\right)dx&amp;=&amp;-\\int\\frac{\\left(2+2\\sin2x\\right)}{\\left(\\cos x-\\sin x\\right)}d\\left(\\cos x-\\sin x\\right)\\\\&amp;=&amp;-2\\int\\frac{\\left(1+\\sin2x\\right){\\displaystyle d}{\\displaystyle\\left(\\cos x-\\sin x\\right)}}{\\left(\\cos x-\\sin x\\right)}\\\\&amp;=&amp;-2\\int\\frac{\\left(1+2\\sin x\\cos x\\right){\\displaystyle d}{\\displaystyle\\left(\\cos x-\\sin x\\right)}}{\\left(\\cos x-\\sin x\\right)}\\\\&amp;=&amp;-2\\int\\frac{\\left[1+1-\\left(\\left(\\cos x-\\sin x\\right)^2\\right)\\right]d\\left(\\cos x-\\sin x\\right)}{\\left(\\cos x-\\sin x\\right)}\\\\&amp;=&amp;-4\\int\\frac{d\\left(\\cos x-\\sin x\\right)}{\\left(\\cos x-\\sin x\\right)}+2\\int\\left(\\cos x-\\sin x\\right)d\\left(\\cos x-\\sin x\\right)\\\\&amp;=&amp;-4\\ln\\left|\\cos x-\\sin x\\right|+\\left(\\cos x-\\sin x\\right)^2+c\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n<p class=\"has-text-align-left\"><strong>\u0db4\u0dc4\u0dad\u0dd2\u0db1\u0dca \u0daf\u0d9a\u0dca\u0dc0\u0dcf \u0d87\u0dad\u0dca\u0dad\u0dda \u0db8\u0dda \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d9c\u0dd0\u0da7\u0dc5\u0dd4 \u0dc0\u0dd2\u0dc3\u0db3\u0dd3\u0db8\u0da7 \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dca \u0dba\u0ddc\u0daf\u0dcf \u0d9c\u0db1\u0dca\u0db1\u0dcf \u0d86\u0d9a\u0dcf\u0dbb\u0dba \u0dc0\u0dda. \u0db8\u0dd9\u0dba \u0dc3\u0db8\u0dca\u0db6\u0db1\u0dca\u0db0\u0dc0 \u0db4\u0dc3\u0dd4\u0dc0 \u0dc0\u0dd2\u0dc3\u0dca\u0dad\u0dbb\u0dcf\u0dad\u0dca\u0db8\u0d9a\u0dc0 \u0dc3\u0dcf\u0d9a\u0dca\u0da0\u0dca\u0da1\u0dcf \u0d9a\u0dd9\u0dbb\u0dd9\u0db1 \u0db1\u0dd2\u0dc3\u0dcf \u0db4\u0dc4\u0dad\u0dd2\u0db1\u0dca \u0dc0\u0dd2\u0dc3\u0db3\u0dd4\u0db8 \u0db4\u0db8\u0dab\u0d9a\u0dca \u0daf\u0d9a\u0dca\u0dc0\u0dcf \u0d87\u0dad.<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d8b\u0daf\u0dcf\u0dc4\u0dbb\u0dab 07<\/li><\/ul>\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\int\\frac{\\sin x+\\sin^3x}{\\cos2x}dx\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}t&amp;=&amp;\\cos x\\Rightarrow\\frac{dt}{dx}=-\\sin x\\Rightarrow dt=-\\sin xdx\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{\\sin x+\\sin^3x}{\\cos2x}dx&amp;=&amp;\\int\\frac{\\sin{\\displaystyle x}{\\displaystyle\\left(1+\\sin^2x\\right)}{\\displaystyle\\left(\\sin xdx\\right)}}{\\displaystyle\\left(2\\cos^2x-1\\right)}\\\\&amp;=&amp;\\int\\frac{\\left(2-t^2\\right){\\displaystyle\\left(-dt\\right)}}{\\left(2t^2-1\\right)}\\\\&amp;=&amp;-2\\int\\frac{dt}{\\left(2t^2-1\\right)}+\\int\\frac{t^2dt}{\\left(2t^2-1\\right)}\\\\&amp;=&amp;-2\\int\\frac{\\displaystyle dt}{\\displaystyle\\left(2t^2-1\\right)}+\\frac12\\int\\frac{t^2}{\\left(t^2-{\\displaystyle\\frac12}\\right)}\\\\&amp;=&amp;-2\\int\\frac{\\displaystyle dt}{\\displaystyle\\left(2t^2-1\\right)}+\\frac12\\int dt+\\frac14\\int\\frac{dt}{\\left(t^2-\\frac12\\right)}\\\\&amp;=&amp;\\frac{\\displaystyle1}{\\displaystyle2}\\int dt-\\frac34\\int\\frac{\\displaystyle dt}{\\displaystyle\\left(t^2-\\frac{\\displaystyle1}{\\displaystyle2}\\right)}\\\\&amp;=&amp;\\frac{\\displaystyle1}{\\displaystyle2}\\int dt-\\frac32\\int\\frac{dt}{\\left(2t^2-1\\right)}\\\\&amp;=&amp;\\frac{\\displaystyle1}{\\displaystyle2}\\int dt-\\frac34\\int\\left[\\frac1{\\displaystyle\\left(\\sqrt2t-1\\right)}-\\frac1{\\left(\\sqrt2t+1\\right)}\\right]dt\\\\&amp;=&amp;\\frac12\\cos x-\\frac3{4\\sqrt2}\\ln\\left|\\frac{\\sqrt2{\\displaystyle\\cos}{\\displaystyle x}{\\displaystyle-}{\\displaystyle1}}{\\sqrt2\\cos x+1}\\right|+c\\\\&amp;&amp;\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n\n\n<p class=\"has-text-align-center has-background\" style=\"background-color: #272062;text-align: center\"><span style=\"font-size: 18pt;color: #ffffff\"><em><strong><span style=\"font-family: 'book antiqua', palatino, serif\">\u201cWholeness is not achieved by cutting off a portion of one\u2019s being, but by integration of the contraries.\u201d<\/span><\/strong><\/em><\/span><br \/><span style=\"font-family: tahoma, arial, helvetica, sans-serif;font-size: 10pt;color: #808080\">-Carl Jung-<\/span><\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n<p>&nbsp;<\/p>\n\n\n\n<div class=\"wp-block-spacer\" style=\"height: 100px\" aria-hidden=\"true\">\u00a0<\/div>\n\n\n\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button is-style-shadow td_btn_normal\"><a class=\"wp-block-button__link\" style=\"border-radius: 15px\" href=\"https:\/\/drive.google.com\/uc?id=1zaYSXmmKtEI7L634oT3ufxFF4YbXDqbX&amp;export=download\" target=\"_blank\" rel=\"noreferrer noopener\">\u0db4\u0dcf\u0da9\u0db8\u0dda \u0dc3\u0da7\u0dc4\u0db1 Download \u0d9a\u0dbb\u0d9c\u0db1\u0dca\u0db1.<\/a><\/div>\r\n<\/div>\n\n\n\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button is-style-shadow td_btn_normal\"><a class=\"wp-block-button__link\" href=\"https:\/\/drive.google.com\/drive\/folders\/1nckWIt5wB-xVw56bK1UE2TWCsXw7jlNl?usp=sharing\" style=\"border-radius:15px\" target=\"_blank\" rel=\"noreferrer noopener\">\u0dad\u0dc0\u0dad\u0dca \u0db4\u0dca\u200d\u0dbb\u0dc1\u0dca\u0db1 \u0db4\u0dd9\u0db1\u0dca\u0dc0\u0db1\u0dca\u0db1.<\/a><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-spacer\" style=\"height: 100px\" aria-hidden=\"true\">\u00a0<\/div>\n\n\n\n<p>&nbsp;<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u0dc3\u0dd4\u0daf\u0dd4\u0dc3\u0dd4 \u0d86\u0daf\u0dda\u0dc1 \u0dba\u0ddc\u0daf\u0dcf \u0dc0\u0dd2\u0da0\u0dbd\u0dca\u200d\u0dba\u0dba \u0db8\u0dcf\u0dbb\u0dd4 \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dd2\u0dbb\u0dd3\u0db8 \u0db8\u0dd9\u0dc4\u0dd2\u0daf\u0dd3  \u0d85\u0db0\u0dca\u200d\u0dba\u0db1\u0dca\u200d\u0dba\u0dba \u0d9a\u0dbb\u0dba\u0dd2.<\/p>\n","protected":false},"author":67,"featured_media":16557,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"tdm_status":"","tdm_grid_status":"","footnotes":""},"categories":[3671,3635,42,3630,3629],"tags":[3701,4004,3698,3699,4006,4005,4002,4001,3703,3702,4003],"class_list":{"0":"post-11672","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-04-05-integration","8":"category-04-calculus","9":"category-advanced-level-science","10":"category-pure-mathematics","11":"category-combined-mathematics","12":"tag-anukalanaya","13":"tag-anuklnaya","14":"tag-calculus","15":"tag-kalanaya","16":"tag-vichalya-maru-kireemen-anukalanaya","17":"tag-vichalya-maru-kirimen-anukalanaya","18":"tag-wichalya-maru-kireemen-anukalanaya-kireema","19":"tag-wichalya-maru-kirimen-anukalanaya-kirima","20":"tag-3703","21":"tag-3702","22":"tag-4003"},"_links":{"self":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11672"}],"collection":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/users\/67"}],"replies":[{"embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/comments?post=11672"}],"version-history":[{"count":38,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11672\/revisions"}],"predecessor-version":[{"id":34726,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11672\/revisions\/34726"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/media\/16557"}],"wp:attachment":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/media?parent=11672"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/categories?post=11672"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/tags?post=11672"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}