{"id":11367,"date":"2021-05-19T01:28:51","date_gmt":"2021-05-18T19:58:51","guid":{"rendered":"https:\/\/learnsteer.sasnaka.org\/science\/?p=11367"},"modified":"2021-10-12T23:24:03","modified_gmt":"2021-10-12T17:54:03","slug":"04-05-03","status":"publish","type":"post","link":"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/combined-mathematics\/04-05-03\/","title":{"rendered":"04.05.03 &#8211; \u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0d9a\u0dcf\u0dbb -2"},"content":{"rendered":"\r\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\r\n<div class=\"wp-block-button is-style-shadow\"><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>\r\n\r\n\r\n\r\n<h4 class=\"has-text-align-center wp-block-heading\"><span class=\"has-inline-color\" style=\"color: #304170\">5.<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{px+q}{\\sqrt{ax^2+bx+c}}dx\\;<\/span>\u0d86\u0d9a\u0dcf\u0dbb\u0dba<\/span><\/h4>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0db8\u0dd9\u0dba \u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0\u0d9a\u0dd2\u0db1\u0dca \u0d85\u0dc0\u0db6\u0ddd\u0db0 \u0d9a\u0dbb \u0d9c\u0db1\u0dd2\u0db8\u0dd4 .<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (01) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{4x+9}{\\sqrt{8\\;-2x\\;-x^2}}dx<\/span><\/p>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0db8\u0dd4\u0dbd\u0dd2\u0db1\u0dca\u0db8 \u0dc4\u0dbb\u0dba\u0dda \u0d87\u0dad\u0dd2 \u0dc0\u0dbb\u0dca\u0d9c\u0db8\u0dd6\u0dbd \u0dc3\u0dc4\u0dd2\u0dad \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1\u0db1\u0dba \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dc5 \u0dba\u0dd4\u0dad\u0dd4\u0dba\u0dd2.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\frac d{dx}\\sqrt{8\\;-2x\\;-x^{2\\;}}=-\\frac{\\left(x+1\\right)}{\\sqrt{8\\;-2x\\;-\\;x^2}}<\/span><\/p>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li><span class=\"wp-katex-eq\" data-display=\"false\">\\sqrt{8\\;-2x\\;-x^{2\\;}\\;}<\/span> \u0dba\u0db1\u0dca\u0db1 \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbd \u0dc0\u0dd2\u0da7 <span class=\"wp-katex-eq\" data-display=\"false\">-\\frac{\\left(x+1\\right)}{\\sqrt{8\\;-2x\\;-\\;x^2}}<\/span> \u0dbd\u0dd9\u0dc3 \u0dbd\u0dd0\u0db6\u0dda.<\/li>\r\n<li>\u0d92 \u0db1\u0dd2\u0dc3\u0dcf <span class=\"wp-katex-eq\" data-display=\"false\">-\\frac{\\left(x+1\\right)}{\\sqrt{8\\;-2x\\;-\\;x^2}}<\/span> \u0dba\u0db1\u0dca\u0db1 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbd \u0dc0\u0dd2\u0da7<span class=\"wp-katex-eq\" data-display=\"false\">\\sqrt{8\\;-2x\\;-x^{2\\;}\\;}<\/span> \u0dbd\u0dd9\u0dc3 \u0dbd\u0dd0\u0db6\u0dd2\u0dba \u0dba\u0dd4\u0dad\u0dd4\u0dba\u0dd2.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\;-\\frac{\\left(x+1\\right)}{\\sqrt{8\\;-2x\\;-\\;x^2}}dx&amp;=&amp;\\sqrt{8\\;-2x\\;-x^{2\\;}\\;}\\\\\\int\\frac{\\left(x+1\\right)}{\\sqrt{8\\;-2x\\;-\\;x^2}}dx&amp;=&amp;-\\sqrt{8\\;-2x\\;-x^{2\\;}\\;}\\end{array}<\/span>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\int\\;\\frac{\\left(x+1\\right)}{\\sqrt{8\\;-2x\\;-\\;x^2}}dx\\end{array}<\/span> \u0dba\u0db1\u0dca\u0db1 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbd \u0dc0\u0dd2\u0da7 \u0dbd\u0dd0\u0db6\u0dd9\u0db1 \u0db4\u0dd2\u0dc5\u0dd2\u0dad\u0dd4\u0dbb \u0d85\u0db4 \u0daf\u0dd0\u0db1\u0dca \u0daf\u0db1\u0dd2\u0db8\u0dd4 .<\/li>\r\n<li>\u0d92 \u0db1\u0dd2\u0dc3\u0dcf <span class=\"wp-katex-eq\" data-display=\"false\">\\frac{4x+9}{\\sqrt{8\\;-2x\\;-x^2}}<\/span> \u0dba\u0db1\u0dca\u0db1 <span class=\"wp-katex-eq\" data-display=\"false\">\\frac{\\left(x+1\\right)}{\\sqrt{8\\;-2x\\;-\\;x^2}}<\/span>\u0d87\u0dc3\u0dd4\u0dbb\u0dd9\u0db1\u0dca \u0dc3\u0d9a\u0dc3\u0dca \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0 \u0dc0\u0dd2\u0dc3\u0daf\u0dcf \u0d9c\u0dad \u0dc4\u0dd0\u0d9a.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\frac{4x+9}{\\sqrt{8\\;-2x\\;-x^2}}=\\frac{4\\left(x+1\\right)+5}{\\sqrt{8\\;-2x\\;-x^2}}<\/span>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{4x+9}{\\sqrt{8\\;-2x\\;-x^2}}dx\\;&amp;=&amp;\\int\\frac{4\\left(x+1\\right)+5}{\\sqrt{8\\;-2x\\;-x^2}}dx\\\\&amp;=&amp;4\\int\\frac{x+1}{\\sqrt{8-2x-x^2}}dx+5\\int\\frac{dx}{\\sqrt{8-2x-x^2}}\\\\&amp;=&amp;-4\\sqrt{8-2x-x^2}+5\\int\\frac{dx}{\\sqrt{3^2-(x+1)^2}}\\\\&amp;=&amp;-4\\sqrt{8-2x-x^2}+5\\sin^{-1}\\left(\\frac{x+1}3\\right)+C\\end{array}<\/span>:C \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<\/p>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (02) <span class=\"wp-katex-eq\" data-display=\"false\">2x+5\\sqrt{12\\;-4x\\;-x^2}dx<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{2x+5}{\\sqrt{12\\;-4x\\;-x^2}}dx<\/span><\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac d{dx}\\sqrt{12\\;-4x\\;-x^2\\;}&amp;=&amp;-\\frac{\\left(x+2\\right)}{\\sqrt{12\\;-4x\\;-x^2}}\\\\\\int\\frac{x+2}{\\sqrt{12-4x-x^2}}dx&amp;=&amp;\\sqrt{12-4x-x^2}\\\\\\int\\frac{2x+5}{\\sqrt{12-4x-x^2}}dx&amp;=&amp;\\int\\frac{2(x+2)+1}{\\sqrt{12-4x-x^2}}dx\\\\&amp;=&amp;2\\int\\frac{x+2}{\\sqrt{12-4x-x^2}}dx+\\int\\frac{dx}{\\sqrt{12-4x-x^2}}\\\\&amp;=&amp;-2\\sqrt{12-4x-x^2}+\\sin^{-1}\\left(\\frac{x+2}4\\right)+C\\end{array}<\/span>:C \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<\/p>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (03) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{4x+3}{\\sqrt{2\\;-x\\;-x^2}}dx<\/span><\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac d{dx}\\sqrt{2\\;-x\\;-x^2}&amp;=&amp;\\frac{-\\left(2x+1\\right)}{2\\sqrt{2\\;-x\\;-x^2}}\\\\\\int\\frac{-(2x+1)}{2\\sqrt{2-x-x^2}}dx&amp;=&amp;\\sqrt{2\\;-x\\;-x^2}\\\\\\int\\frac{(2x+1)}{\\sqrt{2-x-x^2}}dx&amp;=&amp;-2\\sqrt{2\\;-x\\;-x^2}\\\\\\int\\frac{4x+3}{\\sqrt{2-x-x^2}}dx&amp;=&amp;\\int\\frac{2(2x+1)+1}{\\sqrt{2-x-x^2}}dx\\\\&amp;=&amp;2\\int\\frac{(2x+1)}{\\sqrt{2-x-x^2}}dx+\\int\\frac{dx}{\\sqrt{2-x-x^2}}\\\\&amp;=&amp;-4\\sqrt{2-x-x^2}+\\int\\frac{dx}{\\sqrt{\\left({\\displaystyle\\frac32}\\right)^2-\\left(x+{\\displaystyle\\frac12}\\right)^2}}\\\\&amp;=&amp;-4\\sqrt{2-x-x^2}+\\sin^{-1}\\left(\\frac{x+{\\displaystyle\\frac1x}}{\\displaystyle\\frac32}\\right)+C\\end{array}<\/span>:C \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<\/p>\r\n\r\n\r\n\r\n<p><br \/><strong>\u0dc0\u0dd2\u0db7\u0dcf\u0d9c \u0d9c\u0dd0\u0da7\u0dc5\u0dd4<\/strong><br \/>2016<br \/>(15)(a)<br \/>(1)<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{dx}{\\sqrt{3+2x+x^2}}<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<br \/>(2)<span class=\"wp-katex-eq\" data-display=\"false\">\\frac d{dx}\\left(\\sqrt{3+2x+x^2}\\right)<\/span> \u0dc3\u0ddc\u0dba\u0dcf \u0d92 \u0db1\u0dba\u0dd2\u0db1\u0dca <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{x-1}{\\sqrt{3+2x+x^2}}dx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<br \/>\u0d89\u0dc4\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{x+1}{\\sqrt{3+2x+x^2}}dx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<br \/>(1) <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{dx}{\\sqrt{3+2x+x^2}}&amp;=&amp;\\int\\frac{dx}{\\sqrt{4-(x-1)^2}}\\;\\;(10)\\\\&amp;=&amp;\\sin^{-1}\\left(\\frac{x-1}2\\right)+C_1\\end{array}<\/span>:C<sub>1<\/sub> \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<br \/>(2)<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\;\\frac d{dx}\\left(\\sqrt{3+2x+x^2}\\right)&amp;=&amp;\\frac12\\left(3+2x+x^2\\right)^\\frac{-1}2\\times(2-2x)\\;\\;(10)\\\\&amp;=&amp;\\frac{1-x}{\\sqrt{3+2x+x^2}}\\end{array}<\/span><br \/>\u0d92 \u0db1\u0dba\u0dd2\u0db1\u0dca <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{x-1}{\\sqrt{3+2x+x^2}}dx=-\\sqrt{3+2x+x^2}+C_2<\/span>:C<sub>2<\/sub>\u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba (10)<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{x+1}{\\sqrt{3+2x+x^2}}dx&amp;=&amp;\\int\\frac{x-1}{\\sqrt{3+2x+x^2}}dx+2\\int\\frac{dx}{\\sqrt{3+2x+x^2}}(10)\\\\&amp;=&amp;-\\sqrt{3+2x+x^2}+2\\sin^{-1}\\left(\\frac{x-1}2\\right)+C_3\\end{array}<\/span>: C<sub>3<\/sub> \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<br \/><br \/>X \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dc3\u0daf\u0db8\u0dd4.<br \/>-5=2A-B+C (02)<br \/>\u0db1\u0dd2\u0dba\u0dad \u0db4\u0daf \u0dc3\u0db8\u0dcf\u0db1 \u0d9a\u0dbb\u0db8\u0dd4 .<br \/>0=A -C (03)<br \/>( 01),(02),(03) \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab \u0dc0\u0dd2\u0dc3\u0daf\u0dd2\u0db8\u0dd9\u0db1\u0dca ,<br \/>A=-1, B=1,C=-1<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{x^2-5x}{\\left(x-1\\right)\\left(x+1\\right)^2}=\\frac{-1}{\\left(x-1\\right)}+\\frac{x-1}{\\left(x+1\\right)^2}<\/span><\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{x^2-5x}{\\left(x-1\\right)\\left(x+1\\right)^2}dx&amp;=&amp;\\int\\left[\\frac{-1}{\\left(x-1\\right)}\\;+\\frac{x-1}{\\left(x+1\\right)^2}\\right]dx\\\\&amp;=&amp;-\\int\\frac{dx}{(x-1)}+\\int\\frac{x-1}{(x+1)^2}dx\\\\&amp;=&amp;-\\ln\\left|x-1\\right|+\\int\\frac{(x+1)-2}{(x+1)^2}dx\\\\&amp;=&amp;-\\ln\\left|x-1\\right|+\\int\\frac{dx}{(x+1)}-2\\int\\frac{dx}{(x+1)^2}\\\\&amp;=&amp;-\\ln\\left|x-1\\right|+\\ln\\left|x+1\\right|+\\frac2{(x+1)}+C\\end{array}<\/span>:C\u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<\/p>\r\n\r\n\r\n\r\n<p><strong>\u0dc0\u0dd2\u0db7\u0dcf\u0d9c \u0d9c\u0dd0\u0da7\u0dc5\u0dd4<\/strong><br \/>(15)(b)<span class=\"wp-katex-eq\" data-display=\"false\">\\frac{2x-1}{\\left(x+1\\right)\\left(x^2+1\\right)}<\/span> \u0db7\u0dd2\u0db1\u0dca\u0db1 \u0db7\u0dcf\u0d9c \u0d87\u0dc3\u0dd4\u0dbb\u0dd9\u0db1\u0dca \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1 \u0d9a\u0dbb \u0d92 \u0db1\u0dba\u0dd2\u0db1\u0dca<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{2x-1}{\\left(x+1\\right)\\left(x^2+1\\right)}dx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">2x-1\\equiv A\\left(x^2+1\\right)+(Bx\\;+C)\\left(x+1\\right)<\/span> (10)<br \/><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}x^2:0&amp;=&amp;A+B\\\\x^1:2&amp;=&amp;B+C\\\\x^0:1&amp;=&amp;A+C\\end{array}<\/span><br \/>\u0d89\u0dc4\u0dad \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab \u0dc0\u0dd2\u0dc3\u0daf\u0dd2\u0db8\u0dd9\u0db1\u0dca ,<br \/><span class=\"wp-katex-eq\" data-display=\"false\">A=-\\frac32,B=\\frac12,C=\\frac32(10)<\/span><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{2x-1}{\\left(x+1\\right)\\left(x^2+1\\right)}=-\\;\\frac3{2\\left(x+1\\right)}+\\frac{3x+1}{2\\left(x^2+1\\right)}<\/span><\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\"><\/span>:C \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<br \/>(15)<\/p>\r\n\r\n\r\n\r\n<h4 class=\"has-text-align-center wp-block-heading\"><span class=\"has-inline-color\" style=\"color: #304170\">6.\u0db7\u0dd2\u0db1\u0dca\u0db1 \u0db7\u0dcf\u0d9c \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dd2\u0dbb\u0dd3\u0db8<\/span><\/h4>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0db4\u0dbb\u0dd2\u0db8\u0dda\u0dba \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dc3\u0daf\u0dc4\u0dcf \u0db8\u0dd9\u0db8 \u0d9a\u0dca\u200d\u0dbb\u0db8\u0dba \u0dba\u0ddc\u0daf\u0dcf \u0d9c\u0db1\u0dd3 .<\/li>\r\n<li>\u0db4\u0dbb\u0dd2\u0db8\u0dda\u0dba \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba \u0db7\u0dd2\u0db1\u0dca\u0db1 \u0db7\u0dcf\u0d9c\u0dc0\u0dbd\u0da7 \u0dc3\u0dbb\u0dbd \u0d9a\u0dbb \u0d9c\u0dd0\u0db1\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb\u0db1\u0dd4 \u0dbd\u0dd0\u0db6\u0dda .<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p><br \/>\u0d8b\u0daf\u0dcf : (01) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{2x+1}{\\left(x^2+1\\right)\\left(x-2\\right)}dx<\/span><\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac{2x+1}{\\left(x^2+1\\right)\\left(x-2\\right)}&amp;=&amp;\\frac{Ax+B}{x^2+1}+\\frac C{x-2}\\\\&amp;=&amp;\\frac{(Ax+B)(x-2)+C(x^2+1)}{(x^2+1)(x-2)}\\end{array}<\/span>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">2x+1\\equiv\\left(Ax+B\\right)\\left(x-2\\right)+C\\left(x^2+1\\right)<\/span><br \/>x<sup>2<\/sup>\u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dc3\u0daf\u0db8\u0dd4.<br \/>0=A+C (01)<br \/>X \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dc3\u0daf\u0db8\u0dd4<br \/>2=-2A +B (02)<br \/>\u0db1\u0dd2\u0dba\u0dad \u0db4\u0daf \u0dc3\u0db8\u0dcf\u0db1 \u0d9a\u0dbb\u0db8\u0dd4 .<br \/>1=-2B +C (03)<br \/>(01),(02),(03) \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab \u0dc0\u0dd2\u0dc3\u0daf\u0dd2\u0db8\u0dd9\u0db1\u0dca ,<br \/>A=-1, B=0,C=1<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{2x+1}{\\left(x^2+1\\right)\\left(x-2\\right)}=\\frac{-x}{x^2+1}+\\frac1{x-2}<\/span><\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{c}\\int\\frac{2x+1}{\\left(x^2+1\\right)\\left(x-2\\right)}dx=\\int\\left(\\frac{-x}{x^2+1}+\\frac1{x-2}\\right)dx\\\\=\\frac12\\int\\frac{2x}{x^2+1}dx+\\int\\frac{dx}{x-2}\\\\=\\frac12\\ln\\left|x^2+1\\right|+\\ln\\left|x-2\\right|+C\\end{array}<\/span>:c \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<\/p>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (02) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{x^3}{\\left(x-1\\right)\\left(x-2\\right)}dx<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac{x^3}{\\left(x-1\\right)\\left(x-2\\right)}&amp;=&amp;(Ax+B)+\\frac C{(x\\;-1)}+\\;\\frac D{(x\\;-2)}\\\\&amp;=&amp;\\frac{(Ax+B)(x-1)(x-2)+C(x-2)+D(x-1)}{(x-1)(x-2)}\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p>x<sup>3<\/sup> \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dc3\u0daf\u0db8\u0dd4<br \/>1=A (01)<br \/>x<sup>2<\/sup>\u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dc3\u0daf\u0db8\u0dd4.<br \/>0=-3A+B (02)<br \/>X \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dc3\u0daf\u0db8\u0dd4<br \/>0=-3B+2A+C+D (03)<br \/>\u0db1\u0dd2\u0dba\u0dad \u0db4\u0daf \u0dc3\u0db8\u0dcf\u0db1 \u0d9a\u0dbb\u0db8\u0dd4 .<br \/>0=2B -2C -D (04)<br \/>(01),(02),(03),(04) \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab \u0dc0\u0dd2\u0dc3\u0daf\u0dd2\u0db8\u0dd9\u0db1\u0dca ,<br \/>A=1,B=3, C=-1,D=8<\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{x^3}{(x-1)(x-2)}=(x+3)-\\;\\frac1{\\left(x-1\\right)}+\\frac8{\\left(x-2\\right)}<\/span><\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{x^3}{\\left(x-1\\right)\\left(x-2\\right)}dx&amp;=&amp;\\int\\left[(x+3)-\\frac1{(x-1)}+\\frac8{(x-2)}\\right]dx\\\\&amp;=&amp;\\int(x+3)dx-\\int\\frac{dx}{(x-1)}+\\int\\frac{dx}{(x-2)}\\\\&amp;=&amp;\\frac{(x+3)^2}2-\\ln\\left|x-1\\right|+\\ln\\left|x-2\\right|+C\\end{array}<\/span>:C \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<\/p>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (03) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{x^2-5x}{\\left(x-1\\right)\\left(x+1\\right)^2}dx<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac{x^2-5x}{\\left(x-1\\right)\\left(x+1\\right)^2}&amp;=&amp;\\frac A{\\left(x-1\\right)}+\\frac{Bx+C}{\\left(x+1\\right)^2}\\\\&amp;=&amp;\\frac{A(x+1)^2+(Bx+C)(x-1)}{(x-1)(x+1)^2}\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">x^2-5x\\equiv\\;A\\left(x+1\\right)^2+\\left(Bx+C\\right)\\left(x-1\\right)<\/span>\r\n\r\n\r\n\r\n<p><br \/>x<sup>2<\/sup>\u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dc3\u0daf\u0db8\u0dd4.<br \/>1=A+B (01)<br \/>X \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dc3\u0daf\u0db8\u0dd4.<br \/>-5=2A-B+C (02)<br \/>\u0db1\u0dd2\u0dba\u0dad \u0db4\u0daf \u0dc3\u0db8\u0dcf\u0db1 \u0d9a\u0dbb\u0db8\u0dd4 .<br \/>0=A -C (03)<br \/>( 01),(02),(03) \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab \u0dc0\u0dd2\u0dc3\u0daf\u0dd2\u0db8\u0dd9\u0db1\u0dca ,<br \/>A=-1,B=1,C=-1<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{x^2-5x}{\\left(x-1\\right)\\left(x+1\\right)^2}=\\frac{-1}{\\left(x-1\\right)}+\\frac{x-1}{\\left(x+1\\right)^2}<\/span><\/p>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{x^2-5x}{\\left(x-1\\right)\\left(x+1\\right)^2}dx&amp;=&amp;\\int\\left[\\frac{-1}{\\left(x-1\\right)}\\;+\\frac{x-1}{\\left(x+1\\right)^2}\\right]dx\\\\&amp;=&amp;\\int\\frac{dx}{(x-1)}+\\int\\frac{x-1}{(x+1)^2}dx\\\\&amp;=&amp;\\ln\\left|x-1\\right|+\\int\\frac{(x+1)-2}{(x+1)^2}dx\\\\&amp;=&amp;\\ln\\left|x-1\\right|+\\int\\frac{dx}{(x+1)}+2\\int\\frac{dx}{(x+1)^2}+C\\end{array}<\/span>:C \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba<\/p>\r\n\r\n\r\n\r\n<p><strong>\u0dc0\u0dd2\u0db7\u0dcf\u0d9c \u0d9c\u0dd0\u0da7\u0dc5\u0dd4<\/strong><br \/>(15)(b)<span class=\"wp-katex-eq\" data-display=\"false\">\\;\\frac{2x-1}{\\left(x+1\\right)\\left(x^2+1\\right)}<\/span> \u0db7\u0dd2\u0db1\u0dca\u0db1 \u0db7\u0dcf\u0d9c \u0d87\u0dc3\u0dd4\u0dbb\u0dd9\u0db1\u0dca \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1 \u0d9a\u0dbb \u0d92 \u0db1\u0dba\u0dd2\u0db1\u0dca <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{2x-1}{\\left(x+1\\right)\\left(x^2+1\\right)}dx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\frac{2x-1}{\\left(x+1\\right)\\left(x^2+1\\right)}=\\frac A{x+1}+\\frac{Bx\\;+C}{x^2+1}(10)<\/span>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">2x-1\\equiv A\\left(x^2+1\\right)+(Bx\\;+C)\\left(x+1\\right)<\/span><br \/>x<sup>2<\/sup>: 0=A+B<br \/>x<sup>1<\/sup>: 2=B+C<br \/>x<sup>0<\/sup>:-1=A+C<br \/>\u0d89\u0dc4\u0dad \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab \u0dc0\u0dd2\u0dc3\u0daf\u0dd2\u0db8\u0dd9\u0db1\u0dca ,<br \/><span class=\"wp-katex-eq\" data-display=\"false\">A=-\\frac32,B=\\frac12,C=\\frac32(10)<\/span><\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\frac{2x-1}{\\left(x+1\\right)\\left(x^2+1\\right)}=-\\;\\frac3{2\\left(x+1\\right)}+\\frac{3x+1}{2\\left(x^2+1\\right)}<\/span>\r\n\r\n\r\n\r\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{2x-1}{\\left(x+1\\right)\\left(x^2+1\\right)}dx&amp;=&amp;-\\frac32\\;\\int\\frac{dx}{\\left(x+1\\right)}+\\frac12\\int\\frac{3x}{\\left(x^2+1\\right)}dx+\\frac12\\int\\frac{dx}{x^2+1}(05)\\\\&amp;=&amp;-\\frac32\\ln\\left|x+1\\right|+\\frac34\\ln\\left|x^2+1\\right|+\\frac12\\tan^{-1}x+C\\end{array}<\/span>:C \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba(15)<\/p>\r\n\r\n\r\n\r\n<h4 class=\"has-text-align-center wp-block-heading\"><span class=\"has-inline-color\" style=\"color: #304170\">7.<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{a\\cos x+b\\sin x}{c\\cos x+d\\sin x}dx<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba.<\/span><\/h4>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>\u0dc4\u0dbb\u0dba\u0dda \u0dc3\u0dc4 \u0dbd\u0dc0\u0dba\u0dda \u0d87\u0dad\u0dd2 \u0dc3\u0dd1\u0db8 \u0db4\u0daf\u0dba\u0d9a\u0db8 sinx \u0dc4\u0dcf cosx \u0dc0\u0dbd \u0d92\u0d9a\u0da2 \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1 \u0dad\u0dd2\u0db6\u0dda \u0db1\u0db8\u0dca ,\r\n<ol>\r\n<li>\u0db4\u0dc5\u0db8\u0dd4\u0dc0 \u0dc4\u0dbb\u0dba\u0dda \u0d85\u0dc0\u0d9a\u0dbd\u0db1 \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a\u0dba \u0dc3\u0ddc\u0dba\u0dcf\u0d9c\u0db1\u0dca\u0db1<\/li>\r\n<li>\u0d89\u0db1\u0dca\u0db4\u0dc3\u0dd4 ,<span class=\"wp-katex-eq\" data-display=\"false\">\u0dbd\u0dc0\u0dba\\equiv\\lambda\\left(\u0dc4\u0dbb\u0dba\\right)+\\mu\\left(\\text{\u0dc4\u0dbb\u0dba\u0dda\u00a0\u0d85\u0dc0\u0d9a\u0dbd\u0db1\u00a0\u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a\u0dba}\\;\\right)<\/span>\u0dbd\u0dd9\u0dc3 \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1 \u0d9a\u0dbb\u0d9c\u0dd9\u0db1 \u0db8\u0dd9\u0db8 \u0dc3\u0dbb\u0dca\u0dc0\u0dc3\u0dcf\u0db8\u0dca\u200d\u0dba \u0dc0\u0dd2\u0dc3\u0daf\u0dd3\u0db8\u0dd9\u0db1\u0dca <span class=\"wp-katex-eq\" data-display=\"false\">\\lambda<\/span>\u0dc3\u0dc4<span class=\"wp-katex-eq\" data-display=\"false\">\\mu<\/span>\u0dc3\u0ddc\u0dba\u0dcf\u0d9c\u0db1\u0dca\u0db1.<\/li>\r\n<li>. \u0daf\u0dd0\u0db1\u0dca \u0daf\u0dd3 \u0d87\u0dad\u0dd2 \u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0\u0dda \u0dbd\u0dc0\u0dba \u0dc3\u0db3\u0dc4\u0dcf \u0d89\u0dc4\u0dad \u0dc3\u0d9a\u0dc3\u0dca \u0d9a\u0dbb\u0d9c\u0dad\u0dca \u0dbd\u0dc0\u0dba \u0dba\u0dd9\u0daf\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db4\u0dc4\u0dc3\u0dd4\u0dc0\u0dd9\u0db1\u0dca \u0dc3\u0dd2\u0daf\u0dd4\u0d9a\u0dbb\u0d9c\u0dad \u0dc4\u0dd0\u0d9a\u0dd2\u0dba. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (1)<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{cosx\\;+\\;sinx\\;}{2cosx\\;+sinx}dx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac d{dx}\\left(2cosx+sinx\\right)&amp;=&amp;\\;-2sinx+cosx\\\\\\cos x+\\sin x&amp;\\equiv&amp;\\lambda(2\\cos x+\\sin x)+\\mu(-2\\sin x+\\cos x)\\\\\\cos x+\\sin x&amp;\\equiv&amp;(2\\lambda+\\mu)\\cos x+(\\lambda-2\\mu)\\sin x\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>cosx \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dd0\u0dc3\u0db3\u0dd3\u0db8.<br \/>1 = 2<span class=\"wp-katex-eq\" data-display=\"false\">\\lambda<\/span> +<span class=\"wp-katex-eq\" data-display=\"false\"> \\mu<\/span> &#8212;&#8212;&#8212;- 1<br \/>sinx \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dd0\u0dc3\u0db3\u0dd3\u0db8.<br \/>1 = <span class=\"wp-katex-eq\" data-display=\"false\">\\lambda<\/span> &#8211; 2<span class=\"wp-katex-eq\" data-display=\"false\">\\mu<\/span> &#8212;&#8212;&#8212;- 2<br \/>1 \u0dc3\u0dc4 2 \u0db1\u0dca,<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\lambda = \\frac{3}{5} , \\mu = -\\frac{1}{5}<\/span><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf ,<span class=\"wp-katex-eq\" data-display=\"false\"> \\begin{array}{rcl}\\int\\frac{cosx\\;+\\;sinx}{2cosx\\;+sinx}dx&amp;=&amp;\\int\\frac{\\frac35\\left(2cosx\\;+\\;sinx\\right)-\\frac15(-2sinx\\;+cosx)}{2cosx\\;+sinx}dx\\\\&amp;=&amp;\\frac35\\int dx-\\frac15\\int\\frac{(-2sinx\\;+\\;cosx)}{2cosx\\;+\\;sinx}dx\\\\&amp;=&amp;\\frac35x-\\frac15\\ln\\left|2\\cos x+\\sin x\\right|+C\\end{array}]<\/span><\/p>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (2) <span class=\"wp-katex-eq\" data-display=\"false\">\\int{\\frac{sinx}{sinx\\ +\\ cosx}dx}<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac d{dx}(sinx+cosx)&amp;=&amp;cosx-sinx\\\\sinx&amp;\\equiv&amp;\\;\\lambda\\left(sinx+cosx\\right)+\\mu(cosx-sinx)\\\\sinx&amp;\\equiv&amp;\\left(\\lambda-\\mu\\right)sinx+(\\lambda+\\mu)cosx\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>sinx \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dd0\u0dc3\u0db3\u0dd3\u0db8.<br \/>1 = <span class=\"wp-katex-eq\" data-display=\"false\">\\lambda<\/span>&#8211;<span class=\"wp-katex-eq\" data-display=\"false\">\\mu<\/span> &#8212;&#8212;&#8212;- 1<br \/>cosx \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dd0\u0dc3\u0db3\u0dd3\u0db8.<br \/>0 = <span class=\"wp-katex-eq\" data-display=\"false\">\\lambda<\/span>+<span class=\"wp-katex-eq\" data-display=\"false\">\\mu<\/span> &#8212;&#8212;&#8212;- 2<br \/>1 \u0dc3\u0dc4 2 \u0db1\u0dca ,<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\lambda=\\frac{1}{2}\\ , \\mu=-\\frac{1}{2}<\/span><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf , <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{sinx}{sinx\\;+cosx}dx&amp;=&amp;\\int\\frac{\\frac12\\left(sinx\\;+cosx\\right)\\;-\\frac12(cosx\\;-\\;sinx)}{sinx\\;+\\;cosx}dx\\\\&amp;=&amp;\\frac12\\int dx+-\\int\\frac{\\cos x-\\sin x}{\\sin x+\\cos x}dx\\\\&amp;=&amp;\\frac12x-\\frac12\\ln\\left|\\sin x+\\cos x\\right|+C\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (3) <span class=\"wp-katex-eq\" data-display=\"false\">\\int{\\frac{2sin3x\\ -cos3x}{3cos3x\\ +\\ 4sin3x\\ }dx}<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac d{dx}(3cos3x+4sin3x)&amp;=&amp;3\\left(-sin3x\\right).3\\;+4cos3x.3\\\\&amp;=&amp;\\;-9sin3x+12cos3x\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}2sin3x-cos3x&amp;\\equiv&amp;\\;\\lambda\\left(3cos3x\\;+4sin3x\\right)+\\;\\mu(-9sin3x\\;+\\;12cos3x)\\\\2\\sin3x-\\cos3x&amp;\\equiv&amp;(3\\lambda+12\\mu)\\cos3x+(4\\lambda-9\\mu)\\sin3x\\end{array}<\/span>\r\n\r\n\r\n\r\n<p>sin3x \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dd0\u0dc3\u0db3\u0dd3\u0db8.<br \/>2 =4<span class=\"wp-katex-eq\" data-display=\"false\">\\lambda<\/span> &#8211; 9<span class=\"wp-katex-eq\" data-display=\"false\">\\mu<\/span> &#8212;&#8212;&#8212;- 1<br \/>cos3x\\ \u0db4\u0daf\u0dc0\u0dbd \u0dc3\u0d82\u0d9c\u0dd4\u0dab\u0d9a \u0dc3\u0dd0\u0dc3\u0db3\u0dd3\u0db8.<br \/>-1=3<span class=\"wp-katex-eq\" data-display=\"false\">\\lambda<\/span>+12<span class=\"wp-katex-eq\" data-display=\"false\">\\mu<\/span> &#8212;&#8212;&#8212;- 2<br \/>1 \u0dc3\u0dc4 2 \u0db1\u0dca,<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\lambda=\\frac{1}{5}\\ ,\\ \\mu=\\ -\\frac{2\\ }{15}<\/span><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf , <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{2sin3x\\;-cos3x}{3cos3x\\;+4sin3x\\;}dx&amp;=&amp;\\int\\frac{{\\displaystyle\\frac15}\\left(3cos3x\\;+\\;4sin3x\\right)\\;-{\\displaystyle\\frac2{15}}(-9sin3x\\;+12cos3x)}{3cos3x\\;+4sin3x}dx\\\\&amp;=&amp;\\frac15\\int dx-\\frac2{15}\\int\\frac{(-9\\sin3x+12\\cos3x)}{3\\cos3x+4\\sin3x}dx\\\\&amp;=&amp;\\frac x5-\\frac2{15}\\ln\\left|3\\cos3x+4\\sin3x\\right|+C\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<h4 class=\"has-text-align-center wp-block-heading\"><span class=\"has-inline-color\" style=\"color: #304170\">8.<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\sin^mxdx<\/span>\u0dc3\u0dc4 <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\cos^mxdx<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba.<\/span><\/h4>\r\n\r\n\r\n\r\n<ul class=\"wp-block-list\">\r\n<li>m \u0d94\u0dad\u0dca\u0dad\u0dda \u0dc4\u0ddd \u0d89\u0dbb\u0da7\u0dca\u0da7\u0dda \u0dc0\u0dd3\u0db8 \u0d85\u0db1\u0dd4\u0dc0 \u0db8\u0dd9\u0db8 \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d9c\u0dd0\u0da7\u0dc5\u0dd4 \u0d86\u0d9a\u0dcf\u0dbb \u0daf\u0dd9\u0d9a\u0d9a\u0da7 \u0dc0\u0dd2\u0dc3\u0db3\u0dd2\u0dba \u0dc4\u0dd0\u0d9a.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n<h3 class=\"wp-block-heading\"><span class=\"has-inline-color\" style=\"color: #0098da\">m \u0d89\u0dbb\u0da7\u0dca\u0da7\u0dda \u0d85\u0dc0\u0dc3\u0dca\u0dae\u0dcf\u0dc0<\/span><\/h3>\r\n\r\n\r\n\r\n<p><br \/>\u0db8\u0dd9\u0dc0\u0dd2\u0da7<span class=\"wp-katex-eq\" data-display=\"false\">\\cos{2\\theta}<\/span> \u0db4\u0dca\u200d\u0dbb\u0dc3\u0dcf\u0dbb\u0dab\u0dba \u0db8\u0d9c\u0dd2\u0db1\u0dca<span class=\"wp-katex-eq\" data-display=\"false\"> \\sin^m{x}<\/span> \u0dc4\u0ddd <span class=\"wp-katex-eq\" data-display=\"false\">\\cos^m{x}<\/span> \u0db4\u0daf\u0dba \u0dbb\u0dda\u0d9a\u0dd3\u0dba \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1\u0db1\u0dc0\u0dbd\u0da7 \u0dc4\u0dd0\u0dbb\u0dc0\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dc3\u0dd2\u0daf\u0dd4\u0d9a\u0dbd \u0dba\u0dd4\u0dad\u0dd4\u0dba.<\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\cos2\\theta=2\\cos^2\\theta-1\\\\\\cos2\\theta=1-2\\sin^2\\theta\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d8b\u0daf\u0dcf : (1) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\sin^{2\\;}xdx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0dc0\u0dd2\u0dc3\u0db3\u0dd4\u0db8 :<span class=\"wp-katex-eq\" data-display=\"false\"> \\cos{2x=1-2\\sin^2{x}}<\/span> \u0db1\u0dd2\u0dc3\u0dcf,<br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\sin^2{x=\\frac{1}{2}-\\frac{1}{2}\\cos{2x}}<\/span><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf , <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\sin^{2\\;}xdx\\;&amp;=&amp;\\;\\int{(\\frac12-\\frac12\\cos2x)}dx\\\\&amp;=&amp;\\;\\frac12\\int dx-\\frac12\\int cos2xdx\\\\&amp;=&amp;\\;\\frac12x-\\frac14sin2x+c\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d8b\u0daf\u0dcf : (2) <span class=\"wp-katex-eq\" data-display=\"false\">\\int{\\cos^2{4x}dx}<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0dc0\u0dd2\u0dc3\u0db3\u0dd4\u0db8 :<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\cos8x&amp;=&amp;2\\cos^24x-1\\;\\\\cos^24x&amp;=&amp;\\frac12+\\frac12\\cos8x\\end{array}<\/span><br \/><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf ,<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\cos^{2\\;}4x\\;dx&amp;=&amp;\\int{(\\frac12\\;+\\frac12cos8x)dx}\\\\&amp;=&amp;\\frac12\\int dx+\\frac12\\int cos8x\\;dx\\\\&amp;=&amp;\\;\\frac12x+\\frac12.\\frac18sin8x+c\\\\&amp;=&amp;\\;\\frac12x+\\frac1{16}\\;sin8x+c\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d8b\u0daf\u0dcf : (3) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\cos^{4\\ }{3x\\ dx}<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0dc0\u0dd2\u0dc3\u0db3\u0dd4\u0db8 :<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\cos6x\\;\\;&amp;=&amp;\\;2\\cos^23x-1\\\\\\cos^23x&amp;=&amp;\\frac12(\\;1+\\;\\cos{6x\\;)}\\\\cos^43x&amp;=&amp;\\left[\\frac12(1+cos6x)\\rbrack^2\\right]\\\\&amp;=&amp;\\;\\frac14(1+2\\cos6x+\\;\\cos^26x)\\\\&amp;=&amp;\\;\\frac14+\\frac12\\cos6x+\\;\\frac14\\cos^26x\\\\&amp;&amp;\\\\\\cos12x&amp;=&amp;\\;2\\cos^26x-1\\\\&amp;&amp;\\;\\cos^2{6x=\\frac12(\\cos12x+1)}\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf , <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\cos^43x&amp;=&amp;\\frac14\\;+\\frac12\\cos{6x+\\frac14.\\frac12(\\cos12x+1)}\\\\&amp;=&amp;\\frac14\\;+\\frac12\\cos6x+\\frac18\\cos12x+\\frac18\\\\&amp;=&amp;\\;\\frac38+\\frac12\\cos6x+\\frac18\\cos12x\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf , <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\cos^{4\\;}3x\\;dx&amp;=&amp;\\;\\int{(\\frac12\\cos6x+\\frac18\\cos12x+\\frac38)dx\\;}\\\\&amp;=&amp;\\;\\frac12\\int\\cos6xdx+\\frac18\\int\\cos12xdx+\\frac38\\int dx\\\\&amp;=&amp;\\;\\frac12.\\frac16\\sin6x+\\frac18.\\frac1{12}\\sin12x+\\frac38x+\\;c\\\\&amp;=&amp;\\frac1{12}\\sin6x+\\frac1{96}\\sin12x+\\frac38x+c\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<h3 class=\"wp-block-heading\"><span class=\"has-inline-color\" style=\"color: #0098da\">m \u0d94\u0dad\u0dca\u0dad\u0dda \u0d85\u0dc0\u0dc3\u0dca\u0dae\u0dcf\u0dc0.<\/span><\/h3>\r\n\r\n\r\n\r\n<p><span class=\"has-inline-color\" style=\"color: #304170\"><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\left\\{f(x)\\right\\}^nf^1(x)dx=\\frac{\\left\\{f(x)\\right\\}^{n+1}}{n+1}+c<\/span><\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/><br \/>\u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0da7 \u0d85\u0db1\u0dd4\u0dc0 \u0dc3\u0d9a\u0dc3\u0dca \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0db8\u0dd9\u0db8 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dc3\u0dd2\u0daf\u0dd4\u0d9a\u0dc5 \u0dc4\u0dd0\u0d9a.<\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d8b\u0daf\u0dcf : (1) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\sin^3{x}dx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\sin^3x\\;dx&amp;=&amp;\\;\\int\\sin^2x.\\sin x\\;dx\\\\&amp;=&amp;\\;\\int{(1-\\cos^2x)}\\sin x\\;dx\\\\&amp;=&amp;\\;\\int\\sin xdx-\\int\\cos^2x\\sin xdx\\\\&amp;=&amp;\\;\\int\\sin x\\;dx+\\;\\int\\cos^2{x(-\\sin x)}dx\\\\&amp;=&amp;\\;-\\cos x+\\frac13\\cos^3x+c\\;\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d8b\u0daf\u0dcf : (2) <span class=\"wp-katex-eq\" data-display=\"false\">\\int{\\cos^3{2x}dx\\ }<\/span>\u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\cos^{3\\;}2xdx&amp;=&amp;\\int\\cos^22x.\\cos2xdx\\\\&amp;=&amp;\\;\\int\\left(1-\\sin^22x\\right).\\cos2xdx\\\\&amp;=&amp;\\;\\int\\cos2x\\;dx-\\;\\int\\sin^22x.\\cos2xdx\\\\&amp;=&amp;\\;\\int\\cos2x\\;dx-\\frac12\\int\\sin^2{2x(2\\cos{2x)\\;dx}}\\\\&amp;=&amp;\\;\\frac{\\sin2x}2-\\frac12\\frac{\\sin^32x}3+c\\\\&amp;=&amp;\\;\\frac{\\sin2x}2-\\;\\frac{\\sin^32x\\;\\;}6+c\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (3) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\sin^5{x}dx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\sin^{5\\;}x\\;dx&amp;=&amp;\\;\\;\\int\\sin^4x.\\sin xdx\\\\&amp;=&amp;\\int{(1-\\cos^2x)}^2\\sin xdx\\\\&amp;=&amp;\\;\\int\\left(1-2\\cos^2x+\\;\\cos^4x\\right)\\sin xdx\\\\&amp;=&amp;\\;\\int\\sin xdx-2\\int\\cos^2x\\sin x\\;dx+\\;\\int\\cos^4x\\sin xdx\\\\&amp;=&amp;\\;\\int\\sin xdx+2\\int\\cos^2x\\left(-\\sin x\\right)dx-\\;\\int\\cos^4x\\left(-\\sin x\\right)dx\\\\&amp;=&amp;\\;-\\cos x+\\frac23\\cos^3x-\\frac15\\cos^5x+c\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d8b\u0daf\u0dcf : (4)<span class=\"wp-katex-eq\" data-display=\"false\"> \\int\\cos^7{5x}dx<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\cos^75x\\;dx&amp;=&amp;\\int\\cos^65x.\\cos5x\\;dx\\\\&amp;=&amp;\\int{(1-\\sin^25x)}^3.\\cos5xdx\\\\&amp;=&amp;\\;\\int{(1-3\\sin^25x+\\;3\\sin^45x-\\;\\sin^65x)\\cos5x}dx\\\\&amp;=&amp;\\;\\int\\cos5x\\;dx-3\\int\\sin^25x.\\cos5x\\;dx+3\\int\\sin^45x\\cos5xdx-\\;\\int\\sin^65x\\cos5x\\;dx\\\\&amp;=&amp;\\;\\int\\cos5xdx-\\frac35\\int{\\sin^25x.(5\\cos5x)}dx+\\frac35\\int\\sin^45x\\left(5\\cos5x\\right)dx-\\frac15\\int{\\sin^65x(5\\cos5x)}dx\\\\&amp;=&amp;\\;\\frac{\\sin5x}5-\\frac35.\\frac13\\sin^35x+\\frac35.\\frac15\\sin^55x-\\frac15.\\frac17\\sin^75x+c\\\\&amp;=&amp;\\;\\frac{\\sin5x}5-\\frac15\\;\\sin^35x+\\frac3{25}\\sin^55x-\\frac1{35}\\sin^75x+c\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p>\u0d8b\u0daf\u0dcf : (5) <span class=\"wp-katex-eq\" data-display=\"false\">\\int\\sin^6{5x}dx <\/span>\u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0dc0\u0dd2\u0dc3\u0db3\u0dd4\u0db8 : <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\cos10x=1-2\\sin^25x\\;\\\\\\sin^2{5x=\\frac12(1-\\cos10x)}\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf , <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\sin^65x\\;&amp;=&amp;{\\frac12(1-\\;\\cos10x)}^3\\\\&amp;=&amp;\\frac18{(1-\\cos10x)}^3\\\\&amp;=&amp;\\;\\frac18(1-3\\cos10x+3\\cos^210x-\\;\\cos^310x)\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{c}\\cos20x=2\\cos^210x-1\\\\\\cos^2{10x=\\frac12(\\cos20x+1)}\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf , <span class=\"wp-katex-eq\" data-display=\"false\">\\sin^6\\left(5x\\right)=\\frac18{1\\;-\\;3\\cos\\left(10x\\right)+\\frac32(\\cos\\left(20x\\right)+1)-\\cos^3\\left(10x\\right)}<\/span><\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;&amp;=&amp;\\frac18\\int\\left\\{1-3\\cos\\left(10x\\right)+\\frac32(\\cos20x+1)-\\cos^310x\\right\\}dx\\\\&amp;=&amp;\\frac18\\int\\left\\{\\frac52-3\\cos\\left(10x\\right)+\\frac32\\cos20x-\\cos^310x\\right\\}dx\\\\&amp;=&amp;\\frac5{16}\\int dx-\\frac38\\int\\cos\\left(10x\\right)dx+\\frac3{16}\\int\\cos20x\\;dx-\\frac18\\int\\cos^310x\\;dx\\end{array}<\/span>\r\n\r\n\r\n\r\n<p><br \/>I =<span class=\"wp-katex-eq\" data-display=\"false\">\\;\\int\\cos^310x\\;dx\\;\\;\\;<\/span> \u0dba\u0dd0\u0dba\u0dd2 \u0d9c\u0db1\u0dd2\u0db8\u0dd4.<\/p>\r\n\r\n\r\n\r\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}I&amp;=&amp;\\;\\int\\cos^310x\\;dx\\\\&amp;=&amp;\\;\\int\\cos^210x\\;.\\cos10xdx\\\\&amp;=&amp;\\;\\int\\left(1-\\sin^210x\\right).\\cos10xdx\\\\&amp;=&amp;\\;\\int\\cos10xdx-\\int\\sin^210x\\cos10xdx\\\\&amp;=&amp;\\;\\int\\cos10xdx-\\frac1{10}\\int\\sin^2{10x(10\\cos10x)dx}\\\\&amp;=&amp;\\;\\frac{\\sin10x}{10}-\\frac1{10}.\\frac13.\\sin^310x+c\\\\&amp;=&amp;\\;\\frac{\\sin10x}{10}-\\frac1{30}\\sin^310x+c\\end{array}<\/span>\r\n\r\n\r\n\r\n<p><br \/>\u0d91\u0db8\u0db1\u0dd2\u0dc3\u0dcf , <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\;\\int\\sin^65x\\;dx&amp;=&amp;\\frac5{16}x-\\frac38.\\frac1{10}.\\sin10x+\\frac3{16}.\\frac1{20}.\\sin{20x-\\frac18(\\frac{\\sin10x}{10}-\\frac{\\sin^310x}{30})}+c\\\\&amp;=&amp;\\;\\frac5{\\;\\;16}x-\\frac3{80}\\sin10x+\\frac3{320}\\sin20x-\\frac{\\sin10x}{80}+\\frac{\\sin^310x}{240}+c\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0d8b\u0daf\u0dcf : (6)<span class=\"wp-katex-eq\" data-display=\"false\"> \\int{\\sin^3{x\\cos^2{x}}dx}<\/span> \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\r\n\r\n\r\n\r\n<p><br \/>\u0dc0\u0dd2\u0dc3\u0db3\u0dd4\u0db8 : <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\sin^22x\\cos^32xdx&amp;=&amp;\\int\\sin^22x.\\cos^22x.\\cos2xdx\\\\&amp;=&amp;\\;\\int\\sin^22x\\left(1-\\sin^22x\\right).\\cos2x\\;dx\\\\&amp;=&amp;\\int\\sin^22x\\cos2x\\;dx-\\;\\int\\sin^42x\\cos2x\\;dx\\\\&amp;=&amp;\\;\\frac12\\int\\sin^22x\\left(2\\cos2x\\right)dx-\\frac12\\int\\sin^4{2x(2\\cos2x)dx}\\\\&amp;=&amp;\\;\\frac12.\\frac13.\\sin^32x-\\frac12.\\frac15\\;\\sin^52x+c\\\\&amp;=&amp;\\;\\frac16\\sin^32x-\\frac1{10}\\sin^52x+c\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n<div class=\"wp-block-spacer\" style=\"height: 100px\" aria-hidden=\"true\">\u00a0<\/div>\r\n\r\n\r\n\r\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\r\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>\r\n\r\n\r\n\r\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-block-buttons-is-layout-flex\">\r\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\/drive\/folders\/1nckWIt5wB-xVw56bK1UE2TWCsXw7jlNl?usp=sharing\" 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>\r\n<\/div>\r\n\r\n\r\n\r\n<div class=\"wp-block-spacer\" style=\"height: 100px\" aria-hidden=\"true\">\u00a0<\/div>\r\n\r\n\r\n\r\n<p>&nbsp;<\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>\u0d85\u0db1\u0dd2\u0dc1\u0dca\u0da0\u0dd2\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd \u0dc3\u0dd9\u0dc0\u0dd3\u0db8 \u0dc4\u0dcf \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba\u0dda \u0d9c\u0dd0\u0da7\u0dc5\u0dd4 \u0dba\u0dd9\u0daf\u0dd9\u0db1 \u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0d9a\u0dcf\u0dbb \u0db4\u0dd2\u0dc5\u0dd2\u0db6\u0db3\u0dc0 \u0db8\u0dd9\u0dc4\u0dd2 \u0daf\u0dd3 \u0dc0\u0dd0\u0da9\u0dd2\u0daf\u0dd4\u0dbb\u0da7\u0dad\u0dca \u0dc3\u0dcf\u0d9a\u0da0\u0dca\u0da1\u0dcf \u0d9a\u0dbb\u0dba\u0dd2.<\/p>\n","protected":false},"author":58,"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":[3701,3706,3707,3698,3699,3700,3703,3705,3704,3702],"class_list":{"0":"post-11367","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-anukalanaya-bawitha","14":"tag-anukalanaya-bhawitha","15":"tag-calculus","16":"tag-kalanaya","17":"tag-klnaya","18":"tag-3703","19":"tag-3705","20":"tag-3704","21":"tag-3702"},"_links":{"self":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11367"}],"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\/58"}],"replies":[{"embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/comments?post=11367"}],"version-history":[{"count":51,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11367\/revisions"}],"predecessor-version":[{"id":32581,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11367\/revisions\/32581"}],"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=11367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/categories?post=11367"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/tags?post=11367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}