{"id":11692,"date":"2021-05-19T19:04:28","date_gmt":"2021-05-19T13:34:28","guid":{"rendered":"https:\/\/learnsteer.sasnaka.org\/science\/?p=11692"},"modified":"2022-02-22T11:56:29","modified_gmt":"2022-02-22T06:26:29","slug":"04-05-06","status":"publish","type":"post","link":"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/04-05-06\/","title":{"rendered":"04.05.06 &#8211; \u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0daf\u0dda\u0dc1 &#8211; 1"},"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\">\u0d86\u0daf\u0dda\u0dc1 \u0db8\u0d9c\u0dd2\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba<\/h2>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<ul class=\"wp-block-list\"><li>\u0d86\u0daf\u0dda\u0dc1\u0dba:<span class=\"wp-katex-eq\" data-display=\"false\">\\ t=tan({\\frac{x}{2})}<\/span> \u0dba\u0ddc\u0daf\u0dcf \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb\u0db8\u0dd4.<br>\u0d8b\u0daf\u0dcf:<br><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{dx}{2+cosx}\\ <\/span>\u0dc3\u0dbd\u0d9a\u0db8\u0dd4.<br>t=<span class=\"wp-katex-eq\" data-display=\"false\">\\tan{\\left(\\frac{x}{2}\\right)} <\/span>\u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4<br>x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,<\/li><\/ul>\n\n\n\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}dt&amp;=&amp;\\frac12\\sec^2\\frac x2\\;dx\\\\dt&amp;=&amp;\\frac12\\left(1+\\tan^{2\\;}\\frac x2\\right)dx\\\\dt&amp;=&amp;\\frac12\\left(1+t^2\\right)dx\\\\dx&amp;=&amp;\\frac{2dt}{(1+t^2)}\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\cos x&amp;=&amp;\\cos^{2\\;}\\frac x2\\;-\\sin^2\\frac x2\\\\&amp;=&amp;\\frac{\\cos^{2\\;}\\frac x2\\;-\\sin^2\\frac x2}{\\cos^2\\frac x2+\\sin^2\\frac x2}\\end{array}<\/span><br>\u0dc4\u0dbb\u0dba \u0dc4\u0dcf \u0dbd\u0dc0\u0dba <span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\cos^{2\\;}\\frac x2\\end{array}<\/span> \u0db1\u0dca \u0db6\u0dd9\u0daf\u0dd6 \u0dc0\u0dd2\u0da7,<br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\cos x&amp;=&amp;\\;\\frac{1-\\tan^2{\\displaystyle\\frac x2}}{1+\\;\\tan^2\\frac x2}\\\\\\cos x&amp;=&amp;\\;\\frac{1-t^2}{1+t^2}\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{dx}{2+cosx}&amp;=&amp;\\int\\frac{\\frac{2dt}{(1+t^2)}}{2+\\left(\\frac{1-t^2}{1+t^2}\\right)}\\\\&amp;=&amp;\\int\\frac{2dt}{2+2t^2+1-t^2}\\\\&amp;=&amp;2\\int\\frac{dt}{t^2+3}\\\\&amp;=&amp;2\\int\\frac{dt}{t^2+\\left(\\sqrt3\\right)^2}\\\\&amp;=&amp;\\frac2{\\sqrt3}\\tan^{-1}\\left(\\frac t{\\sqrt3}\\right)\\\\&amp;=&amp;\\frac2{\\sqrt3}\\tan^{-1}\\left(\\frac{\\tan\\frac x2}{\\sqrt3}\\right)+c\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong><span class=\"has-inline-color\" style=\"color: #304170\">3.<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{\\mathbf{dx}}{\\mathbf{a}+\\mathbf{bcosx}+\\mathbf{csinx}\\ }<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d86\u0daf\u0dda\u0dc1\u0dba: t=<span class=\"wp-katex-eq\" data-display=\"false\">\\tan{\\left(\\frac{x}{2}\\right)}<\/span> \u0dba\u0ddc\u0daf\u0dcf \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb\u0db8\u0dd4.<br>\u0d8b\u0daf\u0dcf:<br><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{dx}{5+4sinx+3cosx}<\/span> \u0dc3\u0dbd\u0d9a\u0db8\u0dd4.<br>t=<span class=\"wp-katex-eq\" data-display=\"false\">\\tan{\\left(\\frac{x}{2}\\right)}<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.<br>x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,<\/li><\/ul>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong><span class=\"has-inline-color\" style=\"color: #304170\">4.<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{\\mathbf{dx}}{\\mathbf a+\\mathbf b\\mathbf{cos}^{\\mathbf2}\\mathbf x\\;+\\mathbf c\\mathbf{sin}^{\\mathbf2}\\mathbf x\\;}<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd <\/span><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d86\u0daf\u0dda\u0dc1\u0dba:<span class=\"wp-katex-eq\" data-display=\"false\">t=\\tan x<\/span> \u0dba\u0ddc\u0daf\u0dcf \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb\u0db8\u0dd4.<br>\u0d8b\u0daf\u0dcf:<br><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{dx}{3\\cos^2x-4\\sin^2x-5}<\/span> \u0dc3\u0dbd\u0d9a\u0db8\u0dd4.<br><span class=\"wp-katex-eq\" data-display=\"false\">t=\\tan x<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.<br>x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca<\/li><\/ul>\n\n\n\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}dt&amp;=&amp;\\sec^2xdx\\\\dt&amp;=&amp;\\left(1+\\tan^{2\\;}x\\right)dx\\\\dt&amp;=&amp;\\left(1+t^2\\right)dx\\\\dx&amp;=&amp;\\frac{dt}{(1+t^2)}\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\sin^2x&amp;=&amp;\\frac1{\\mathrm c\\mathrm o\\mathrm s\\mathrm e\\mathrm c^2x}\\\\\\sin^2x&amp;=&amp;\\frac1{1+\\cot^2x}\\\\\\sin^2x&amp;=&amp;\\frac{\\tan^2x}{1+\\tan^2x}\\\\\\sin^2x&amp;=&amp;\\frac{t^2}{1+t^2}\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\cos^2x&amp;=&amp;\\frac1{\\sec^2x}\\\\\\cos^2x&amp;=&amp;\\frac1{1+\\tan^2x}\\\\\\cos^2x&amp;=&amp;\\frac1{1+t^2}\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{dx}{3\\cos^2\\;x-4\\sin^2x-5}&amp;=&amp;\\int\\frac{\\frac{dt}{(1+t^2)}}{3\\left(\\frac1{1+t^2}\\right)-4\\left(\\frac{t^2}{1+t^2}\\right)-5}\\\\&amp;=&amp;\\int\\frac{2dt}{3-4t^2-5-5t^2}\\\\&amp;=&amp;\\int\\frac{dt}{-9t^2-2}\\\\&amp;=&amp;-\\int\\frac{dt}{9t^2+2}\\\\&amp;=&amp;-\\frac19\\int\\frac{dt}{t^2+\\frac29}\\\\&amp;=&amp;-\\frac19\\int\\frac{dt}{t^2+\\left(\\frac{\\sqrt2}3\\right)^2}\\\\&amp;=&amp;-\\frac19.\\frac3{\\sqrt2}\\tan^{-1}\\left(\\frac t{\\displaystyle\\frac{\\sqrt2}3}\\right)+c\\\\&amp;=&amp;-\\frac1{3\\sqrt2}\\tan^{-1}\\left(\\frac3{\\sqrt2}\\tan x\\right)+c\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<div class=\"wp-block-cover has-background-dim\" style=\"background-color:#b1cbdf\"><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<p class=\"has-text-color\" style=\"color:#38556d;font-size:20px\">&nbsp;<strong><span class=\"wp-katex-eq\" data-display=\"false\">x\\geq1<\/span> \u0dc3\u0db3\u0dc4\u0dcf <span class=\"wp-katex-eq\" data-display=\"false\">y=\\frac1x<\/span> \u0dc0\u0d9a\u0dca\u200d\u0dbb\u0dba x \u0d85\u0d9a\u0dca\u0dc2\u0dba \u0dc0\u0da7\u0dcf \u0db7\u0dca\u200d\u0dbb\u0db8\u0dab\u0dba \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0dc3\u0dd1\u0daf\u0dd9\u0db1 \u0d9d\u0dab \u0dc0\u0dc3\u0dca\u0dad\u0dd4\u0dc0 \u0dc3\u0dbd\u0d9a\u0db1\u0dca\u0db1.<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"535\" height=\"233\" src=\"https:\/\/learnsteer.sasnaka.org\/science\/wp-content\/uploads\/sites\/3\/2021\/11\/04.05.06_img_Gaberiels_horn.jpg\" alt=\"\" class=\"wp-image-34789\" \/><\/figure><\/div>\n\n\n\n<p class=\"has-text-color\" style=\"color:#38556d;font-size:20px\"><strong>1.\u0d91\u0db8 \u0d9d\u0dab \u0dc0\u0dc3\u0dca\u0dad\u0dd4\u0dc0\u0dda \u0db4\u0dbb\u0dd2\u0db8\u0dcf\u0dc0 \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.( <span class=\"wp-katex-eq\" data-display=\"false\">y=f(x)<\/span> \u0dc0\u0db1 \u0d9d\u0dd8\u0dab \u0db1\u0ddc\u0dc0\u0db1 \u0dc0\u0d9a\u0dca\u200d\u0dbb\u0dba\u0d9a\u0dca x \u0d85\u0d9a\u0dca\u0dc2\u0dba \u0dc0\u0da7\u0dcf \u0db7\u0dca\u200d\u0dbb\u0db8\u0dab\u0dba \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0dc3\u0dd1\u0daf\u0dd9\u0db1 \u0d9d\u0dab \u0dc0\u0dc3\u0dca\u0dad\u0dd4\u0dc0\u0dda \u0db4\u0dbb\u0dd2\u0db8\u0dcf\u0dc0 \u0db8\u0dd9\u0db8 \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab\u0dba\u0dd9\u0db1\u0dca \u0dbd\u0db6\u0dcf \u0daf\u0dda. <span class=\"wp-katex-eq\" data-display=\"false\">v=\\mathrm\\pi\\int_{\\mathrm a}^{\\mathrm b}{\\mathrm f{(\\mathrm x)}^2}\\;\\mathrm{dx}\\;;\\;\\mathrm a\\leq\\mathrm x\\leq\\mathrm b<\/span> )<\/strong><\/p>\n\n\n\n<p class=\"has-text-color\" style=\"color:#38556d;font-size:20px\"><strong>2.<span class=\"wp-katex-eq\" data-display=\"false\">y=f(x)<\/span> \u0dc0\u0db1 \u0d9d\u0dd8\u0dab \u0db1\u0ddc\u0dc0\u0db1 \u0dc0\u0d9a\u0dca\u200d\u0dbb\u0dba\u0d9a\u0dca x \u0d85\u0d9a\u0dca\u0dc2\u0dba \u0dc0\u0da7\u0dcf \u0db7\u0dca\u200d\u0dbb\u0db8\u0dab\u0dba \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0dc3\u0dd1\u0daf\u0dd9\u0db1 \u0d9d\u0dab \u0dc0\u0dc3\u0dca\u0dad\u0dd4\u0dc0\u0d9a \u0db8\u0dad\u0dd4\u0db4\u0dd2\u0da7 \u0dc0\u0dbb\u0dca\u0d9c\u0db5\u0dbd\u0dba \u0db4\u0dc4\u0dad \u0dc3\u0db8\u0dd3\u0d9a\u0dbb\u0dab\u0dba\u0dd9\u0db1\u0dca \u0dbd\u0db6\u0dcf \u0daf\u0dda.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-left has-text-color\" style=\"color:#38556d;font-size:20px\"><strong><span class=\"wp-katex-eq\" data-display=\"false\">A=2\\mathrm\\pi\\int_{\\mathrm a}^{\\mathrm b}\\mathrm f(\\mathrm x)\\;\\sqrt{1+{{\\mathrm f^\/(\\mathrm x)}}^2}\\;\\mathrm{dx}\\;;\\;\\mathrm a\\leq\\mathrm x\\leq\\mathrm b<\/span><\/strong><br><strong>\u0d89\u0dc4\u0dad \u0daf\u0dd0\u0d9a\u0dca\u0dc0\u0dd9\u0db1\u0dca \u0d9d\u0dab \u0dc0\u0dc3\u0dca\u0dad\u0dd4\u0dc0\u0da7 \u0d85\u0dc3\u0dd3\u0db8\u0dd2\u0dad \u0dc0\u0dbb\u0dca\u0d9c\u0db5\u0dbd\u0dba\u0d9a\u0dca \u0dc4\u0dcf \u0dc3\u0dd3\u0db8\u0dd2\u0dad \u0db4\u0dbb\u0dd2\u0db8\u0dcf\u0dc0\u0d9a\u0dca \u0d87\u0dad\u0dd2 \u0db6\u0dc0 \u0db4\u0dd9\u0db1\u0dca\u0dc0\u0db1\u0dca\u0db1.<\/strong><\/p>\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\"><a class=\"wp-block-button__link has-background\" href=\"https:\/\/learnsteer.sasnaka.org\/science\/?p=34791\" style=\"border-radius:10px;background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(56,85,109) 0%)\" target=\"_blank\" rel=\"noreferrer noopener\">\u0db4\u0dd2\u0dc5\u0dd2\u0dad\u0dd4\u0dbb \u0db4\u0dd9\u0db1\u0dca\u0dc0\u0db1\u0dca\u0db1<\/a><\/div>\n<\/div>\n\n\n\n<p><\/p>\n<\/div><\/div>\n\n\n\n<p><strong><span class=\"has-inline-color\" style=\"color: #304170\">5.<span class=\"wp-katex-eq\" data-display=\"false\"> \\int\\frac{\\mathbf{dx}}{\\left(\\mathbf{px}+\\mathbf{q}\\right)\\sqrt{\\mathbf{ax}+\\mathbf{b}}\\; }<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li style=\"text-align: left\">\u0d86\u0daf\u0dda\u0dc1\u0dba:<span class=\"wp-katex-eq\" data-display=\"false\">\\;t=\\sqrt{ax+b}<\/span> \u0dba\u0ddc\u0daf\u0dcf \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb\u0db8\u0dd4.<br>\u0d8b\u0daf\u0dcf:<br><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{dx}{(x+2)\\sqrt{x+1}}<\/span>\u0dc3\u0dbd\u0d9a\u0db8\u0dd4.<br><span class=\"wp-katex-eq\" data-display=\"false\">\\;t=\\sqrt{x+1}<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.<br>x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,<\/li><\/ul>\n\n\n\n<p><\/p>\n\n\n\n<p><strong><span class=\"has-inline-color\" style=\"color: #304170\">6.<span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{\\mathbf{dx}}{\\left(\\mathbf{px}+\\mathbf q\\right)\\sqrt{\\mathbf{ax}^{\\mathbf2}+\\mathbf{bx}+\\mathbf c}\\;\\;}<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u0d86\u0daf\u0dda\u0dc1\u0dba:<span class=\"wp-katex-eq\" data-display=\"false\">\\left(px+q\\right)=\\frac1t<\/span> \u0dba\u0ddc\u0daf\u0dcf \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb\u0db8\u0dd4.<br>\u0d8b\u0daf\u0dcf:<br><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\frac{dx}{\\left(x+2\\right)\\sqrt{x^2+x-1}}<\/span>\u0dc3\u0dbd\u0d9a\u0db8\u0dd4.<span class=\"wp-katex-eq\" data-display=\"false\">x+1=\\frac1t<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.<br><span class=\"wp-katex-eq\" data-display=\"false\">x=\\frac1t-1<\/span><\/li><\/ul>\n\n\n\n<p><br>t \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,<br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\frac{dx}{dt}=-\\frac1{t^2}\\\\dx=-\\frac1{t^2}dt\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\frac{dx}{\\left(x+2\\right)\\sqrt{x^2+x-1}}&amp;=&amp;\\int\\frac{-\\frac1{t^{2\\;}}dt}{\\frac1t\\sqrt{\\left(\\frac1t-1\\right)^2+\\left(\\frac1t-1\\right)-1}}\\\\&amp;=&amp;\\int\\frac{dt}{\\frac1t\\sqrt{\\frac1{t^{2\\;}}-\\frac2t+1+\\frac1t-2}}\\\\&amp;=&amp;-\\int\\frac{dt}{t\\sqrt{\\frac{1-2t-t^2+t}{t^2}}}\\\\&amp;=&amp;-\\int\\frac{dt}{t.\\frac1t\\sqrt{1-t-t^2}}\\\\&amp;=&amp;-\\int\\frac{dt}{\\sqrt{(-)(t^2+t-1)}}\\\\&amp;=&amp;-\\int\\frac{dt}{\\sqrt{(-)\\left[\\left(t+\\frac12\\right)^2-1-\\frac14\\right]}}\\\\&amp;=&amp;-\\int\\frac{dt}{\\sqrt{\\left(\\frac{\\sqrt5}2\\right)^2-\\left(t+\\frac12\\right)^2}}\\\\&amp;=&amp;-\\sin^{-1}\\left(\\frac{t+\\frac12}{\\displaystyle\\frac{\\sqrt5}2}\\right)+c\\\\&amp;=&amp;-\\sin^{-1}\\left(\\frac{\\frac1{\\left(x+1\\right)}+\\frac12}{\\frac{\\sqrt5}2}\\right)+c\\\\&amp;=&amp;-\\sin^{-1}\\left(\\frac{2+x+1}{\\sqrt5(x+1)}\\right)+c\\\\&amp;=&amp;-\\sin^{-1}\\left(\\frac{x+3}{\\sqrt5(x+1)}\\right)+c\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong><span class=\"has-inline-color\" style=\"color: #304170\">7.<span class=\"wp-katex-eq\" data-display=\"false\">\\int{\\mathbf{a}^\\mathbf{x}\\mathbf{dx}}<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\n\n\n\n<p>\u0db8\u0dd9\u0dc4\u0dd2 a \u0dba\u0db1\u0dd4 \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2. \u0db8\u0dd9\u0dc0\u0dd0\u0db1\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1 \u0dc3\u0daf\u0dc4\u0dcf t=a^x \u0dc4\u0ddd k \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dca \u0dc0\u0dd2\u0da7 a=e^k \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db1\u0dd4 \u0dbd\u0dd0\u0db6\u0dda.<br>a)<span class=\"wp-katex-eq\" data-display=\"false\">t=a^x<\/span>\u0d86\u0daf\u0dda\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d89\u0dc4\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dc3\u0ddc\u0dba\u0db8\u0dd4.<br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}t&amp;=&amp;a^x\\\\\\ln t&amp;=&amp;ln\\;a^x\\\\\\ln t&amp;=&amp;x\\;ln\\;a\\end{array}<\/span><br>x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,<br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac1t.\\frac{dt}{dx}&amp;=&amp;\\;ln\\;a\\\\\\frac1{t\\ln a}.dt&amp;=&amp;dx\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int a^xdx\\;&amp;=&amp;\\int t.\\frac1{t.lna}dt\\\\&amp;=&amp;\\int\\frac1{\\ln a}dt\\\\&amp;=&amp;\\frac1{\\ln a}\\int dt\\\\&amp;=&amp;\\frac1{\\ln a}.t+c\\\\&amp;=&amp;\\frac1{\\ln a}.a^x+c\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><br>b)<span class=\"wp-katex-eq\" data-display=\"false\"> a=e^k<\/span> \u0d86\u0daf\u0dda\u0dc1\u0dba\u0dd9\u0db1\u0dca<span class=\"wp-katex-eq\" data-display=\"false\">\\;\\int a^xdx\\;<\/span>\u0dc3\u0ddc\u0dba\u0db8\u0dd4.<br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}a&amp;=&amp;e^k\\\\k&amp;=&amp;\\ln\\;a\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int a^xdx\\;&amp;=&amp;\\int e^{kx}dx\\;\\\\&amp;=&amp;\\frac1k\\;e^{kx}+c\\\\&amp;=&amp;\\frac1{\\ln a}.a^x+c\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong><span class=\"has-inline-color\" style=\"color: #304170\">8.<span class=\"wp-katex-eq\" data-display=\"false\">\\;\\int\\sqrt{\\mathbf a^{\\mathbf2}-\\mathbf x^{\\mathbf2}\\;}\\mathbf{dx}<\/span>\u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u0db8\u0dd9\u0dc0\u0dd0\u0db1\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1 <span class=\"wp-katex-eq\" data-display=\"false\">x=a\\sin\\theta<\/span> \u0dc4\u0ddd <span class=\"wp-katex-eq\" data-display=\"false\">x=a\\cos\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0d9a\u0dbb\u0db8\u0dd4.<br>\u0d8b\u0daf\u0dcf.<br><span class=\"wp-katex-eq\" data-display=\"false\">\\int\\sqrt{4-x^2}dx<\/span>\u0dc3\u0dbd\u0d9a\u0db8\u0dd4.<br><span class=\"wp-katex-eq\" data-display=\"false\">x=2\\sin\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.<br>x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,<\/li><\/ul>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}dx\\;&amp;=&amp;\\;2\\;\\cos\\theta\\;d\\theta\\\\\\int\\sqrt{4-x^2}dx&amp;=&amp;\\int\\sqrt{4-4\\sin^2\\theta}.2\\cos\\theta d\\theta\\\\&amp;=&amp;\\int\\sqrt{4(1-\\sin^2\\theta)}.2\\cos\\theta d\\theta\\\\&amp;=&amp;\\int\\sqrt{4\\cos^2\\theta}.2\\cos\\theta d\\theta\\;;\\;{1-\\sin^2\\theta=\\cos^2\\theta}\\\\&amp;=&amp;\\int2\\cos\\theta.2\\cos\\theta\\;d\\theta\\\\&amp;=&amp;2\\int2\\cos^2\\theta\\;d\\theta\\\\&amp;=&amp;2\\int(1+\\cos2\\theta)d\\theta\\;;\\;{\\cos2\\theta=2\\cos^2\\theta-1}\\\\&amp;=&amp;2\\left\\{\\int d\\theta\\;+\\;\\int\\cos2\\theta\\;d\\theta\\right\\}\\\\&amp;=&amp;2\\left\\{\\theta+\\frac12\\sin2\\theta\\right\\}+c\\\\&amp;=&amp;2\\theta+\\sin2\\theta+c\\\\&amp;=&amp;2\\sin^{-1}\\left(\\frac x2\\right)+\\frac x2.\\sqrt{4-x^2}+c\\;(c\\;\u0d85\u0db7\u0dd2\u0db8\u0dad\\;\u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2)\\\\&amp;&amp;\\end{array}<\/span>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\left\\{\\begin{array}{rcl}\\sin\\theta&amp;=&amp;\\frac x2\\\\\\cos\\theta&amp;=&amp;\\sqrt{1-\\sin^2\\theta}\\\\\\cos\\theta&amp;=&amp;\\sqrt{1-\\frac{x^2}4}=\\frac{\\sqrt{4-x^2}}2\\\\\\sin2\\theta&amp;=&amp;2\\sin\\theta\\cos\\theta=2.\\left(\\frac x2\\right).\\left(\\frac{\\sqrt{4-x^2}}2\\right)\\\\\\sin2\\theta&amp;=&amp;\\frac x2.\\sqrt{4-x^2}\\end{array}\\right\\}<\/span>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-center has-background\" style=\"background-color:#272062\"><p class=\"has-text-align-center has-background\" style=\"background-color: #272062;text-align: center\"><span style=\"font-size: 18pt;color: #ffffff\"><strong><span style=\"font-family: 'book antiqua', palatino, serif\">\u201cScience is the Differential Calculus of the mind. Art the Integral Calculus; they may be beautiful when apart, but are greatest only when combined.\u201d<\/span><\/strong><\/span><br><span style=\"font-family: tahoma, arial, helvetica, sans-serif;font-size: 10pt;color: #808080\">-Ronald Ross &#8211;<\/span><\/p><\/p>\n\n\n\n<p>&nbsp;<\/p>\n<\/div><\/div>\n\n\n<p><!-- \/wp:group --><\/p>\n<div class=\"wp-block-group\">\n<div class=\"wp-block-group__inner-container\"><!-- wp:group -->\n<p>\u00a0<\/p>\n<div class=\"wp-block-group\">\n<div class=\"wp-block-group__inner-container\"><!-- wp:paragraph -->\n<p>\u00a0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:paragraph --><!-- \/wp:paragraph --><\/p>\n<p><!-- wp:spacer --><\/p>\n<div class=\"wp-block-spacer\" style=\"height: 100px\" aria-hidden=\"true\">\u00a0<\/div>\n<p><!-- \/wp:spacer --><\/p>\n<p><!-- wp:buttons {\"contentJustification\":\"center\"} --><\/p>\n<div class=\"wp-block-buttons is-content-justification-center\"><!-- wp:button {\"borderRadius\":15,\"className\":\"is-style-shadow\",\"tdButtonSize\":\"normal\"} -->\n<p>\u00a0<\/p>\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\"><span style=\"color: #ffffff\">\u0db4\u0dcf\u0da9\u0db8\u0dda \u0dc3\u0da7\u0dc4\u0db1 Download \u0d9a\u0dbb\u0d9c\u0db1\u0dca\u0db1.<\/span><\/a><\/div>\n<\/div>\n<p><!-- \/wp:button --><\/p>\n<p><!-- wp:buttons {\"contentJustification\":\"center\"} --><\/p>\n<div class=\"wp-block-buttons is-content-justification-center\"><!-- wp:button {\"borderRadius\":15,\"className\":\"is-style-shadow\",\"tdButtonSize\":\"normal\"} -->\n<p>\u00a0<\/p>\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\"><span style=\"color: #ffffff\">\u0dad\u0dc0\u0dad\u0dca \u0db4\u0dca\u200d\u0dbb\u0dc1\u0dca\u0db1 \u0db4\u0dd9\u0db1\u0dca\u0dc0\u0db1\u0dca\u0db1.<\/span><\/a><\/div>\n<p><!-- \/wp:button --><\/p>\n<\/div>\n<p><!-- \/wp:buttons --><\/p>\n<p><!-- wp:spacer --><\/p>\n<div class=\"wp-block-spacer\" style=\"height: 100px\" aria-hidden=\"true\">\u00a0<\/div>\n<p><!-- \/wp:spacer --><\/p>\n<p><!-- wp:paragraph --><\/p>\n<p>\u00a0<\/p>\n<p><!-- \/wp:paragraph --><!-- \/wp:buttons --><!-- \/wp:group --><!-- \/wp:group --><!-- \/wp:group --><!-- \/wp:group --><!-- \/wp:group --><\/p>","protected":false},"excerpt":{"rendered":"<p>\u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0daf\u0dda\u0dc1 1 \u0d9a\u0ddc\u0da7\u0dc3\u0dda\u0daf\u0dd3 \u0dc0\u0dd2\u0dc0\u0dd2\u0db0 \u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0d9a\u0dcf\u0dbb\u0dc0\u0dbd \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9c\u0dd0\u0da7\u0dc5\u0dd4 \u0dc0\u0dd2\u0dc3\u0db3\u0dd3\u0db8\u0da7 \u0d85\u0daf\u0dcf\u0dc5 \u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0daf\u0dda\u0dc1 \u0db4\u0dd2\u0dc5\u0dd2\u0db6\u0db3\u0dc0\u0dad\u0dca \u0d92\u0dc0\u0dcf \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0d8b\u0daf\u0dcf\u0dc4\u0dbb\u0dab \u0d9c\u0dd0\u0da7\u0dc5\u0dd4 \u0dc0\u0dd2\u0dc3\u0db3\u0db1 \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dad\u0dca \u0db4\u0dd0\u0dc4\u0dd0\u0daf\u0dd2\u0dbd\u0dd2 \u0d9a\u0dbb \u0d87\u0dad<\/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":[3701,3706,3707,3699,3703,3705,3704],"class_list":{"0":"post-11692","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-kalanaya","16":"tag-3703","17":"tag-3705","18":"tag-3704"},"_links":{"self":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11692"}],"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=11692"}],"version-history":[{"count":77,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11692\/revisions"}],"predecessor-version":[{"id":35721,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/11692\/revisions\/35721"}],"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=11692"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/categories?post=11692"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/tags?post=11692"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}