{"id":12030,"date":"2021-03-02T10:05:55","date_gmt":"2021-03-02T04:35:55","guid":{"rendered":"https:\/\/learnsteer.sasnaka.org\/science\/?p=12030"},"modified":"2021-11-12T07:57:10","modified_gmt":"2021-11-12T02:27:10","slug":"04-05-05","status":"publish","type":"post","link":"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/combined-mathematics\/04-05-05\/","title":{"rendered":"04.05.05 &#8211; \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba"},"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<p>&nbsp;U \u0dc4\u0dcf V \u0dba\u0db1\u0dd4 x \u0dc4\u0dd2 x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbd \u0dc4\u0dd0\u0d9a\u0dd2 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad \u0daf\u0dd9\u0d9a\u0d9a\u0dca \u0dc0\u0db1 \u0dc0\u0dd2\u0da7,<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"has-inline-color has-vivid-cyan-blue-color\" style=\"color: #000000\"><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{\\operatorname d(UV)}{\\operatorname dx}=U\\frac{\\operatorname dV}{\\operatorname dx}+V\\frac{\\operatorname dU}{\\operatorname dx}<\/span><\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"has-inline-color has-vivid-cyan-blue-color\" style=\"color: #000000\"><span class=\"wp-katex-eq\" data-display=\"false\">U\\frac{\\operatorname dV}{\\operatorname dx}=\\frac{\\operatorname d(UV)}{\\operatorname dx}-V\\frac{\\operatorname dU}{\\operatorname dx}<\/span><\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"has-inline-color has-vivid-cyan-blue-color\" style=\"color: #000000\"><span class=\"wp-katex-eq\" data-display=\"false\">\\int U\\frac{\\operatorname dV}{\\operatorname dx}dx=\\int\\frac{\\operatorname d(UV)}{\\operatorname dx}dx-\\int V\\frac{\\operatorname dU}{\\operatorname dx}Udx<\/span><\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"has-inline-color has-vivid-cyan-blue-color\" style=\"color: #000000\"><span class=\"wp-katex-eq\" data-display=\"false\">\\int U\\frac{\\operatorname dV}{\\operatorname dx}dx=UV-\\int V\\frac{\\operatorname dU}{\\operatorname dx}dx<\/span><\/span><\/p>\n\n\n\n<ul>\r\n<li>\u0d89\u0dc4\u0dad \u0daf\u0d9a\u0dca\u0dc0\u0dcf \u0d87\u0dad\u0dd2 \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0daf\u0d9a\u0dca\u0dc0\u0db1 \u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb\u0dba\u0dda \u0dc0\u0db8\u0dca\u0db4\u0dc3\u0da7 \u0d85\u0db1\u0dd4\u0d9a\u0dd6\u0dbd \u0dc0\u0db1 \u0db4\u0dbb\u0dd2\u0daf\u0dd2 \u0db8\u0dd9\u0db8 \u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0 \u0db4\u0dc4\u0dad \u0daf\u0dd0\u0d9a\u0dca\u0dc0\u0dd9\u0db1 \u0db4\u0dbb\u0dd2\u0daf\u0dd2 \u0d9a\u0dca\u200d\u0dbb\u0db8 \u0daf\u0dd9\u0d9a\u0d9a\u0da7 \u0dc3\u0d9a\u0dc3\u0dca \u0d9a\u0dbd \u0dc4\u0dd0\u0d9a\u0dd2\u0dba.<\/li>\r\n<\/ul>\n\n\n\n<p>\u0283 (\u0db4\u0dc5\u0db8\u0dd4 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba)(\u0daf\u0dd9\u0dc0\u0db1 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba)dx = \u0283 (\u0daf\u0dd9\u0dc0\u0db1 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba)(\u0db4\u0dc5\u0db8\u0dd4 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba\u0dd9 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0dc4\u0dd2 \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba)dx<\/p>\n\n\n\n<p>\u0283 (\u0db4\u0dc5\u0db8\u0dd4 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba)(\u0daf\u0dd9\u0dc0\u0db1 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba)dx = \u0283 (\u0db4\u0dc5\u0db8\u0dd4 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba)(\u0daf\u0dd9\u0dc0\u0db1 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba\u0dd9 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0dc4\u0dd2 \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba)dx<\/p>\n\n\n\n<p>\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 01<\/p>\n\n\n\n<ul>\r\n<li>\u0283 xe<sup>x<\/sup> dx \u0dc4\u0dd2 \u0d89\u0dc4\u0dad \u0d86\u0d9a\u0dd8\u0dad\u0dd2\u0dba\u0da7 \u0d85\u0db1\u0dd4\u0dc0 \u0dc3\u0d9a\u0dc3\u0dca \u0d9a\u0dbd \u0dc0\u0dd2\u0da7,<\/li>\r\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\" style=\"text-align: left\"><span class=\"wp-katex-eq\" data-display=\"false\">\\int xe^xdx=\\int x\\frac{\\operatorname de^x}{\\operatorname dx}dx<\/span><\/p>\n\n\n\n<p>\u0dbd\u0dd9\u0dc3 \u0dc4\u0ddd<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\int xe^xdx=\\int e^x\\frac{\\operatorname d({\\displaystyle\\frac{x^2}2})}{\\operatorname dx}dx<\/span>\n\n\n\n<p>\u0dbd\u0dd9\u0dc3 \u0dc0\u0dda.<\/p>\n\n\n\n<ul>\r\n<li>\u0daf\u0dd0\u0db1\u0dca \u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb\u0dba\u0da7 \u0d85\u0db1\u0dd4\u0dc0 U \u0dc4\u0dcf V \u0db4\u0dc4\u0dc3\u0dd4\u0dc0\u0dd9\u0db1\u0dca \u0dc4\u0daf\u0dd4\u0db1\u0dcf\u0d9c\u0dad \u0dc4\u0dd0\u0d9a\u0dd2 \u0dc0\u0db1 \u0d85\u0dad\u0dbb \u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb\u0dba\u0dda \u0daf\u0d9a\u0dd4\u0dab\u0dd4 \u0db4\u0dc3\u0da7 \u0d85\u0db1\u0dd4\u0d9a\u0dd6\u0dbd \u0dc0\u0db1 \u0db4\u0dbb\u0dd2\u0daf\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dc3\u0dd2\u0daf\u0dd4\u0d9a\u0dbd \u0dc4\u0dd0\u0d9a\u0dd2\u0dba.<\/li>\r\n<\/ul>\n\n\n\n<p>\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-1 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int xe^xdx&amp;=&amp;\\int x\\bullet\\frac{\\operatorname de^x}{\\operatorname dx}dx\\\\&amp;=&amp;xe^x-\\int e^x\\bullet\\frac{\\operatorname dx}{\\operatorname dx}dx\\\\&amp;=&amp;xe^x-\\int e^xdx\\\\&amp;=&amp;xe^x-e^x+C\\end{array}<\/span>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int xe^xdx&amp;=&amp;\\int e^x\\bullet\\frac{\\operatorname d({{\\displaystyle\\frac{x^2}2})}}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac{x^2}2\\bullet e^x-\\int\\frac{x^2}2\\frac{\\operatorname de^x}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac{x^2}2\\bullet e^x-\\int\\frac{x^2}2e^xdx\\end{array}<\/span>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\text{\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(\u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0\u00a0\u0dc3\u0d82\u0d9a\u0dd3\u0dbb\u0dca\u0dab\u00a0\u0dc0\u0dda)}<\/span>\n<\/div>\n<\/div>\n\n\n\n<p>&nbsp;<\/p>\n\n\n\n<p>\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 02<\/p>\n\n\n\n<p>\u0283 x<sup>2<\/sup>cosx dx \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<p>\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<ul>\r\n<li>\u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb\u0dba\u0dda \u0dc0\u0db8\u0dca\u0db4\u0dc3\u0da7 \u0d85\u0db1\u0dd4\u0d9a\u0dd6\u0dbd \u0dc0\u0db1 \u0db4\u0dbb\u0dd2\u0daf\u0dd2 \u0d9a\u0dca\u200d\u0dbb\u0db8 \u0daf\u0dd9\u0d9a\u0d9a\u0da7 \u0db8\u0dd9\u0db8 \u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0 \u0dc3\u0d9a\u0dc3\u0dca \u0d9a\u0dbd \u0dc4\u0dd0\u0d9a\u0dd2\u0dba.<\/li>\r\n<\/ul>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\int x^2\\cos x\\;dx=\\int x^2\\frac{\\operatorname d(\\sin x)}{\\operatorname dx}dx\\;\\text{\u0dbd\u0dd9\u0dc3 \u0dc4\u0ddd}\\int x^2\\cos x\\;dx=\\int\\cos x\\frac{\\operatorname d({\\displaystyle\\frac{x^3}3})}{\\operatorname dx}\\;dx<\/span>\n\n\n\n<p>\u0dbd\u0dd9\u0dc3 \u0dc0\u0dda.<\/p>\n\n\n\n<ul>\r\n<li>\u00a0\u0db8\u0dd9\u0db8 \u0daf\u0dd9\u0dc0\u0db1 \u0d9a\u0dca\u200d\u0dbb\u0db8\u0dba\u0da7 \u0dc3\u0d9a\u0dc3\u0dca \u0d9a\u0dbd \u0dc0\u0dd2\u0da7 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dc3\u0d82\u0d9a\u0dd3\u0dbb\u0dca\u0dab \u0dc0\u0db1 \u0db6\u0dd0\u0dc0\u0dd2\u0db1\u0dca \u0db4\u0dc5\u0db8\u0dd4 \u0dc3\u0d9a\u0dc3\u0dca \u0d9a\u0dd2\u0dbb\u0dd3\u0db8\u0da7 \u0d85\u0db1\u0dd4\u0dc0 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb\u0db8\u0dd4.<\/li>\r\n<\/ul>\n\n\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int x^2\\cos x\\;dx&amp;=&amp;\\int x^2\\frac{\\operatorname d(\\sin x)}{\\operatorname dx}dx\\\\&amp;=&amp;x^2\\sin x\\;-\\;\\int2x\\sin x\\;dx\\\\&amp;=&amp;x^2\\sin x-2\\int x\\frac{\\operatorname d{(-\\cos x)}}{\\operatorname dx}dx\\;\\\\&amp;=&amp;x^2\\sin x-2{-x\\cos x-\\int1\u22c5(-\\cos x)dx}\\;\\\\&amp;=&amp;x^2\\sin x+2x\\cos x-2\\int\\cos xdx\\;\\\\&amp;=&amp;x^2\\sin x+2x\\cos x-2\\sin x+c\\;\u200b\\;\\;\\;\\end{array}<\/span>\n\n\n\n<p>\u0dc3\u0da7\u0dc4\u0db1\u0dca<\/p>\n\n\n\n<ul>\r\n<li>\u0dc3\u0dbb\u0dbd\u0dc0 \u0d9c\u0dad\u0dc4\u0ddc\u0dad\u0dca \u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0\u0d9a\u0dca \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0dc0\u0dd2\u0dc3\u0daf\u0dd3\u0db8\u0dda\u0daf\u0dd2 \u0d91\u0dba\u0da7 \u0d89\u0dc4\u0dad \u0dc3\u0dd6\u0dad\u0dca\u200d\u0dbb\u0dba \u0dba\u0dd9\u0daf\u0dd4 \u0dc0\u0dd2\u0da7 \u0dbd\u0dd0\u0db6\u0dd9\u0db1 <span class=\"wp-katex-eq\" data-display=\"false\">\\int V\\;\\frac{\\operatorname dU}{\\operatorname dx}\\;dx<\/span> \u00a0\u00a0 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbd \u0dc4\u0dd0\u0d9a\u0dd2\u0dc0\u0db1 \u0dbd\u0dd9\u0dc3 U \u0dad\u0ddd\u0dbb\u0dcf\u0d9c\u0dad \u0dba\u0dd4\u0dad\u0dd4\u0dba.<\/li>\r\n<\/ul>\n\n\n\n<p><em>\u0d9c\u0dd0\u0da7\u0dbd\u0dd4 \u0dc0\u0dd2\u0dc3\u0daf\u0dd3\u0db8\u0dda\u0daf\u0dd2 <\/em><em>U \u0dad\u0ddd\u0dbb\u0dcf\u0d9c\u0dad \u0dc4\u0dd0\u0d9a\u0dd2 \u0dc3\u0dbb\u0dbd \u0d9a\u0dca\u200d\u0dbb\u0db8\u0dba\u0d9a\u0dca&#8230;.<\/em><\/p>\n\n\n\n<p><em>I&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; L&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; A&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; T&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; E<\/em><\/p>\n\n\n\n<p><em>I &nbsp;:- \u0db4\u0dca\u200d\u0dbb\u0dad\u0dd2\u0dbd\u0d9d\u0dd4 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad&nbsp;&nbsp; L :- \u0dbd\u0d9d\u0dd4\u0d9c\u0dab\u0d9a \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad&nbsp; A :- \u0dc0\u0dd3\u0da2\u0dd3\u0dba \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad&nbsp; &nbsp;T :- \u0dad\u0dca\u200d\u0dbb\u0dd2\u0d9a\u0ddd\u0dab\u0db8\u0dd2\u0dad\u0dd2\u0d9a \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad&nbsp;&nbsp; E :- \u0d9d\u0dcf\u0dad\u0dd3\u0dba \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad<\/em><\/p>\n\n\n\n<ul>\r\n<li><em>\u0d9c\u0dd0\u0da7\u0dbd\u0dd4\u0dc0\u0d9a \u0dba\u0dd9\u0daf\u0dd3 \u0d87\u0dad\u0dd2 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad \u0daf\u0dd9\u0d9a \u0d89\u0dc4\u0dad <\/em><em>I LATE \u0d85\u0db1\u0dd4\u0db4\u0dd2\u0dc5\u0dd2\u0dc0\u0dd9\u0dbd\u0dd2\u0db1\u0dca \u0db6\u0dd0\u0dbd\u0dd4 \u0dc0\u0dd2\u0da7 \u0db8\u0dd4\u0dbd\u0dd2\u0db1\u0dca\u0db8 \u0dba\u0dd9\u0daf\u0dd9\u0db1 \u0dc1\u0dca\u200d\u0dbb\u0dd2\u0dad\u0dba U \u0dbd\u0dd9\u0dc3 \u0dad\u0ddd\u0dbb\u0dcf\u0d9c\u0dad \u0dc4\u0dd0\u0d9a.<\/em><\/li>\r\n<\/ul>\n\n\n\n<p>\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 03<\/p>\n\n\n\n<p>\u0283 e<sup>x <\/sup>cosx dx &nbsp;\u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<p>\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int e^xcosxdx\\;&amp;=&amp;\\int cosx\\frac{\\operatorname d(e^x)}{\\operatorname dx}dx\\\\&amp;=&amp;e^xcosx-\\int(-sinx)\\bullet e^xdx\\\\&amp;=&amp;e^xcosx+\\int sinx\\frac{\\operatorname de^x}{\\operatorname dx}dx\\\\&amp;=&amp;e^xcosx+e^xsinx-\\int e^xcosxdx\\end{array}<\/span>\n\n\n\n<ul>\r\n<li>\u00a0\u0db8\u0dd9\u0dc4\u0dd2\u0daf\u0dd2 \u0dc0\u0db8\u0dca \u0db4\u0dc3 \u0d87\u0dad\u0dd2 \u0db4\u0daf\u0dba\u0db8 \u0daf\u0d9a\u0dd4\u0dab\u0dd4 \u0db4\u0dc3 \u0dc3\u0dd1\u0daf\u0dd2 \u0d87\u0dad.\u0d91\u0db8 \u0db1\u0dd2\u0dc3\u0dcf \u0d91\u0db8 \u0db4\u0daf\u0dba \u0dc0\u0db8\u0dca \u0db4\u0dc3\u0da7 \u0d9c\u0dd9\u0db1 \u0d86 \u0dba\u0dd4\u0dad\u0dd4\u0dba. \u0d91\u0dc0\u0dd2\u0da7,<\/li>\r\n<\/ul>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}2\\int e^xcosxdx&amp;=&amp;e^x(cosx+sinx)\\\\ e^xcosxdx&amp;=&amp;\\frac12e^x(cosx+sinx)+C\\end{array}<\/span>\n\n\n\n<p>\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 04<\/p>\n\n\n\n<p>\u0283 sin<sup>-1<\/sup>x dx&nbsp; \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<p>&nbsp;\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\\\ sin^{-1}xdx&amp;=&amp;\\ sin^{-1}x\\frac{\\operatorname dx}{\\operatorname dx}dx\\\\&amp;=&amp;xsin^{-1}x-\\int\\frac1{\\sqrt{(1-x^2)}}\\bullet xdx\\\\&amp;=&amp;xsin^{-1}x+\\int\\frac{({\\displaystyle\\frac{-1}2})(-2x)}{\\sqrt{(1-x^2)}}dx\\\\&amp;=&amp;xsin^{-1}x+\\sqrt{1-x^2}+C\\end{array}<\/span>\n\n\n\n\n\n<p>\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 05<\/p>\n\n\n\n<p>\u0283 tan<sup>-1<\/sup>x dx&nbsp; \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<p>&nbsp;\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int tan^{-1}xdx&amp;=&amp;\\int tan^{-1}x\\bullet\\frac{\\operatorname dx}{\\operatorname dx}dx\\\\&amp;=&amp;xtan^{-1}x-\\int\\frac1{1+x^2}\\bullet xdx\\\\&amp;=&amp;xtan^{-1}x\u2013\\frac12\\int\\frac{2x}{1+x^2}dx\\\\&amp;=&amp;xtan^{-1}x\u2013\\frac12ln\\vert1+x^2\\vert+C\\end{array}<\/span>\n\n\n\n\n\n<p>\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 06<\/p>\n\n\n\n<p>\u0283 lnx dx&nbsp; \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<p>&nbsp;\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int\\ln x\\;dx\\;&amp;=&amp;\\;\\int\\ln x\\;\\bullet\\frac{\\operatorname d\\mathrm x}{\\operatorname d\\mathrm x}\\;\\;dx\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\&amp;=&amp;\\;x\\ln x\\;-\\;\\int\\frac{1\\;\\;}x\\bullet x\\;dx\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\&amp;=&amp;\\;x\\ln x\\;-\\int(1)\\;\\;dx\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\&amp;=&amp;\\;x\\ln x\\;\u2013\\;x\\;+\\;C\\end{array}<\/span>\n\n\n\n<p>\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 07<\/p>\n\n\n\n<p>&nbsp;\u0283 x<sup>3<\/sup> lnx dx&nbsp;&nbsp;&nbsp; \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int x^3lnxdx&amp;=&amp;\\int lnx\\bullet\\frac{\\operatorname d({\\displaystyle\\frac{x^4}4})}{\\operatorname d{}}dx\\\\&amp;=&amp;\\frac{x^4}4lnx-\\int\\frac1x\\bullet\\frac{x^4}4dx\\\\&amp;=&amp;\\frac{x^4}4lnx\u2013\\frac14\\int x^3dx\\\\&amp;=&amp;\\frac{x^4}4lnx-\\frac1{16}x^4+C\\end{array}<\/span>\n\n\n\n<p>\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<p>&nbsp;\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 08<\/p>\n\n\n\n<p>\u0283 x e<sup>3x<\/sup>dx&nbsp;&nbsp; \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<p>\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int xe^3xdx&amp;=&amp;\\int x\\frac{\\operatorname d{(\\frac{e^{3x}}3)}}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac{xe^{3x}}3-\\int\\frac{e^{3x}}3dx\\\\&amp;=&amp;\\frac{xe^{3x}}3\u2013\\frac13\\bullet\\frac13e^{3x}+C\\\\&amp;=&amp;\\frac{xe^{3x}}3\u2013\\frac{e^{3x}}9+C\\end{array}<\/span>\n\n\n\n<p>&nbsp;\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1 09<\/p>\n\n\n\n<p>\u0283 e<sup>x<\/sup>sin2x dx&nbsp; \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<p>\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int e^xsin2xdx&amp;=&amp;\\int sin2x\\frac{\\operatorname de^x}{\\operatorname dx}dx\\\\&amp;=&amp;e^xsin2x-2\\int cos2xe^xdx\\\\&amp;=&amp;e^xsin2x-2\\int cos2x\\frac{\\operatorname de^x}{\\operatorname dx}dx\\\\&amp;=&amp;e^xsin2x-2{{e^xcos2x-2\\int e^x(-sin2x)dx}}\\\\&amp;=&amp;e^xsin2x\u20132e^xcos2x-4\\int e^xsin2xdx\\end{array}<\/span>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}5\\int e^xsin2xdx&amp;=&amp;e^x(sin2x-2cos2x)\\\\\\int e^xsin2xdx&amp;=&amp;\\frac{e^x}5(sin2x\u20132cos2x)+C\\end{array}<\/span>\n\n\n\n<p>\u0db1\u0dd2\u0daf\u0dc3\u0dd4\u0db1&nbsp; 10<\/p>\n\n\n\n<p>\u0283 e<sup>2x<\/sup>cosx dx&nbsp; &nbsp;\u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<p>\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int e^2xcosxdx&amp;=&amp;\\int cosx\\frac{\\operatorname d{(\\frac{e^{2x}}2)}}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac12e^{2x}cosx-\\int\\frac12(-sinx)e^{2x}dx\\\\&amp;=&amp;\\frac12e^{2x}cosx+\\frac12\\int sinx\\frac{\\operatorname d{(\\frac12e^{2x}})}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac12e^{2x}cosx+\\frac12{\\frac12e^2xsinx-\\int\\frac12e^{2x}cosxdx}\\\\&amp;=&amp;\\frac12e^{2x}cosx+\\frac14e^{2x}sinx-\\frac14\\int e^{2x}cosxdx\\\\\\frac54\\int e^2xcosxdx&amp;=&amp;{\\frac12e^{2x}cosx+\\frac14e^{2x}sinx}\\\\\\int e^2xcosxdx&amp;=&amp;\\frac45{2e^{2x}cosx+e^{2x}sinx}+C\\end{array}<\/span>\n\n\n\n<div class=\"wp-block-cover has-background-dim\" style=\"background-color:#f3ca58;min-height:510px;aspect-ratio:unset;\"><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<p class=\"has-text-color\" style=\"color:#6d5922;font-size:30px\">   <strong>\u0db4\u0dd4\u0d82\u0da0\u0dd2 \u0d85\u0db7\u0dd2\u0dba\u0ddd\u0d9c\u0dba\u0d9a\u0dca ..<\/strong><\/p>\n\n\n\n<p class=\"has-text-color\" style=\"color:#6d5922;font-size:20px\">   <strong>\u0db8\u0dda \u0dbb\u0dd6\u0db4\u0dba\u0dda \u0dad\u0dd2\u0dba\u0dd9\u0db1 \u0d85\u0db3\u0dd4\u0dbb\u0dd4 \u0d9a\u0dc5 \u0d9a\u0ddc\u0da7\u0dc3\u0dda \u0dc0\u0dbb\u0dca\u0d9c\u0db5\u0dbd\u0dba \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.\u0db8\u0dd9\u0dc4\u0dd2 \u0daf\u0dd0\u0d9a\u0dca\u0dc0\u0dd9\u0db1 \u0d9a\u0dd4\u0da9\u0dcf \u0dc0\u0dd8\u0dad\u0dca\u0dad\u0dba\u0da7 \u0d85\u0daf\u0dcf\u0dc5 \u0dc3\u0dca\u0db4\u0dbb\u0dca\u0dc1\u0d9a\u0dba\u0dda \u0daf\u0dd2\u0d9c \u0d92\u0d9a\u0d9a 8 \u0d9a\u0dd2.<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"220\" height=\"220\" src=\"https:\/\/learnsteer.sasnaka.org\/science\/wp-content\/uploads\/sites\/3\/2021\/11\/image.png\" alt=\"\" class=\"wp-image-34712\" \/><\/figure><\/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\"><a class=\"wp-block-button__link has-background\" href=\"https:\/\/learnsteer.sasnaka.org\/science\/?p=34713\" style=\"border-radius:10px;background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(109,89,34) 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<h4 class=\"wp-block-heading\"><strong>\u0d85\u0db7\u0dca\u200d\u0dba\u0dcf\u0dc3 \u0db8\u0dcf\u0dbd\u0dcf\u0dc0<\/strong><\/h4>\n\n\n\n<p>x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca&nbsp; \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb\u0db1\u0dca\u0db1.<\/p>\n\n\n\n<ol>\r\n<li>x<sup>2<\/sup> e<sup>4x<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/li>\r\n<\/ol>\n\n\n\n<p>2. x<sup>3<\/sup>(lnx)<sup>2<\/sup>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p>3. e<sup>-x<\/sup> sinx &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p>4. x sin(x+\u03c0\/6)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p>5. x cos(nx)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p>6. x<sup>n<\/sup>lnx&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p>7. x sec<sup>2<\/sup>x<\/p>\n\n\n\n<p>8. x<sup>3 <\/sup>tan<sup>-1<\/sup>x&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p>9. e<sup>ax<\/sup> sin(bx)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/p>\n\n\n\n<p>10. x (1+x)<sup>7<\/sup><\/p>\n\n\n\n<p>\u0dc0\u0dd2\u0dc3\u0daf\u0dd4\u0db8\u0dca<\/p>\n\n\n\n<p>1.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int x^2e^4xdx&amp;=&amp;\\int x^2\\frac{\\operatorname d{(\\frac14e^{4x})}}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac14e^{4x}-\\frac14\\int2xe^{4x}dx\\\\&amp;=&amp;\\frac14e^{4x}-\\frac12\\int x\\frac14e^{4x}dx\\\\&amp;=&amp;\\frac14e^{4x}-\\frac12{\\frac14xe^{4x}-\\frac14\\int e^{4x}dx}\\\\&amp;=&amp;\\frac14e^{4x}-\\frac18xe^{4x}+\\frac18\\int xe^{4x}dx\\\\&amp;=&amp;\\frac14e^{4x}-\\frac18xe^{4x}+\\frac1{32}e^{4x}+C\\end{array}<\/span>\n\n\n\n<p>2.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int x^3(lnx)^2dx&amp;=&amp;\\int(lnx)^2\\frac{\\operatorname d{({\\displaystyle\\frac14}x^4)}}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac14x^4(lnx)^2-\\int2(lnx)\\bullet\\frac1x\\bullet\\frac14x^4dx\\\\&amp;=&amp;{\\frac14x^4(lnx)^2}-\\frac12\\int(lnx)x^3dx\\\\&amp;=&amp;{\\frac14x^4(lnx)^2}-\\frac12\\int(lnx)\\frac{\\operatorname d{(\\frac14x^4)}}{\\operatorname dx}dx\\\\&amp;=&amp;{\\frac14x^4(lnx)^2}-\\frac12\\{\\frac14x^4lnx-\\int\\frac12\\bullet\\frac14x^4dx\\\\&amp;=&amp;{\\frac14x^4(lnx)^2}-\\frac18x^4\\ln x+\\frac18\\int x^3dx\\\\&amp;=&amp;{\\frac14x^4(lnx)^2}-\\frac18x^4\\ln x+\\frac1{32}x^4+C\\end{array}<\/span>\n\n\n\n<p>3.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int e(-x)sinxdx&amp;=&amp;\\int sinx\\frac{\\operatorname d{(-e^{-x})}}{\\operatorname dx}dx\\\\&amp;=&amp;-e^{-x}sinx-\\int cosx(-e^{-x})dx\\\\&amp;=&amp;-e^{-x}sinx+\\int e^{-x}cosxdx\\\\&amp;=&amp;-e^{-x}sinx+\\int cosx\\frac{\\operatorname d{(-e^{-x})}}{\\operatorname dx}dx\\\\&amp;=&amp;-e^{-x}sinx-e^{-x}\\cos x-\\int(-sinx)(-e^{-x})dx\\\\&amp;=&amp;-e^{-x}sinx-e^{-x}\\cos x-\\int e^{-x}{sinx}dx\\\\2\\int e^{-x}sinxdx\\;&amp;=&amp;-e^{-x}sinx-e^{-x}\\cos x\\\\\\int e^{-x}sinxdx\\;&amp;=&amp;-e^{-x}\\frac12(sinx+\\cos x)\\end{array}<\/span>\n\n\n\n<p>4.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int xsin(x+\\frac\\pi6)dx&amp;=&amp;\\int x\\frac{\\operatorname d{-\\cos{(x+\\frac\\pi6)}}}{\\operatorname dx}dx\\\\&amp;=&amp;-\\cos{(x+\\frac\\pi6)}-\\int{{-\\cos{(x+\\frac\\pi6)}}}dx\\\\&amp;=&amp;-\\cos{(x+\\frac\\pi6)}+\\int\\cos{(x+\\frac\\pi6)}dx\\\\&amp;=&amp;-\\cos{(x+\\frac\\pi6)}+\\int\\sin{(x+\\frac\\pi6)}dx+C\\end{array}<\/span>\n\n\n\n<p>5.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int xcos(nx)dx&amp;=&amp;\\int x\\frac{\\operatorname d{{{\\displaystyle\\frac1n}\\sin(nx)}}}{\\operatorname dx}dx\\\\&amp;=&amp;{\\frac1n\\sin(nx)}-\\frac1n\\int sin(nx)dx\\\\&amp;=&amp;\\frac xnsin(nx)-\\frac1n(-\\frac1ncosnx)+C\\\\&amp;=&amp;\\frac xnsinnx+\\frac x{n^2}cosnx+C\\end{array}<\/span>\n\n\n\n<p>6.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int x^nlnxdx\\;&amp;=&amp;\\int lnx\\frac{\\operatorname d(\\frac1{n+1}x{}^{n+1})}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac1{n+1}x^{n+1}lnx-\\int\\frac1{n+1}x{}^{n+1}\\frac1x\\\\&amp;=&amp;\\frac1{n+1}x{}^{n+1}lnx-\\int\\frac1{n+1}\\int x^ndx\\\\&amp;=&amp;\\frac1{n+1}x{}^{n+1}lnx-\\frac1{(n+1)^2}x{}^{n+1}\\;+C\\;\\end{array}<\/span>\n\n\n\n<p>7.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int xsec^2xdx&amp;=&amp;\\int x\\;\\frac{\\operatorname d(tanx)}{\\operatorname dx}dx\\\\&amp;=&amp;xtanx-\\int tanxdx\\\\&amp;=&amp;xtanx+ln\\vert cosx\\vert+C\\end{array}<\/span>\n\n\n\n<p>8.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int x^3tan^{-1}xdx&amp;=&amp;\\int tan^{-1}x\\frac{\\operatorname d({\\displaystyle\\frac14}x^4)}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac14x^4tan^{-1}x-\\frac14\\int\\frac{x^4}{x^2+1}x^2+1dx\\end{array}<\/span>\n\n\n\n<p>\u0daf\u0dd3\u0dbb\u0dca\u0d9d \u0db6\u0dd9\u0daf\u0dd3\u0db8\u0dd9\u0db1\u0dca<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\frac{x^4}{x^2+1}&amp;=&amp;x^2\u20131+\\frac1{x^2+1}\\\\\\int x^3tan^{-1}xdx\\;&amp;=&amp;\\frac14x^4tan^{-1}x-\\frac14\\int\\{{x^2\u20131+\\frac1{x^2+1}\\;\\;}\\;dx\\\\&amp;=&amp;\\frac14x^4tan^{-1}x-\\frac1{12}x^3+\\frac x4-\\frac14tan^{-1}x+C\\end{array}<\/span>\n\n\n\n<p>9.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int e^{ax}sin(bx)dx&amp;=&amp;\\int sin(bx)\\frac{\\operatorname d{({\\displaystyle\\frac1a}e^{ax})}}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac1ae^{ax}sinbx-\\frac ba\\int cosbxe^{ax}dx\\\\&amp;=&amp;\\frac1ae^{ax}sinbx-\\frac ba\\int cosbx\\frac{\\operatorname d{({\\displaystyle\\frac1a}e^{ax})}}{\\operatorname dx}dx\\\\&amp;=&amp;\\frac1ae^{ax}sinbx-\\frac ba{\\;{\\frac1a\\;e^{ax}\\cos bx-\\frac ba\\int e^{ax}(-\\sin bx)dx\\;}\\;\\;}\\\\&amp;=&amp;\\frac1ae^{ax}sinbx-\\frac b{a^2}{\\;e^{ax}\\cos bx-\\frac{b^2}{a^2}\\int e^{ax}(-\\sin bx)dx\\;\\;\\;}\\\\(a^2+b^2){\\int e^{ax}\\sin bxdx\\;\\;}&amp;=&amp;e^{ax}(asinbx-bcosbx)\\\\\\int e^{ax}sinbxdx&amp;=&amp;\\frac{e^{ax}}{a^2+b^2}(asinbx\u2013bcosbx)+C\\end{array}<\/span>\n\n\n\n<p>10.<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}x(1+x)^7dx&amp;=&amp;\\int x\\bullet\\frac{\\operatorname d{({\\displaystyle\\frac18}(1+x)^8})}{\\operatorname dx}dx\\\\&amp;=&amp;{{\\frac x8(1+x)^8}\\;}-\\frac18\\int(1+x)^8dx\\\\&amp;=&amp;{\\frac x8(1+x)^8}\\;-\\frac1{72}(1+x)^9+C\\end{array}<\/span>\n\n\n\n<p><strong>\u0dc0\u0dd2\u0db7\u0dcf\u0d9c \u0d9c\u0dd0\u0da7\u0dbd\u0dd4<\/strong><\/p>\n\n\n\n<p>&nbsp;1. \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dba\u0ddc\u0daf\u0d9c\u0db1\u0dd2\u0db8\u0dd2\u0db1\u0dca&nbsp;&nbsp; \u0283 e<sup>3x<\/sup>cos4x dx \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (2004)<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int e^{3x}cos4xdx&amp;=&amp;\\int cos4x\\frac{\\operatorname d({\\displaystyle\\frac13}e^{3x})}{\\operatorname dx}dx\\;\\boxed5\\\\&amp;=&amp;\\frac13e^{3x}cos4x-\\frac13\\int(-4sin4x)e^{3x}dx\\;\\;\\;\\boxed5\\\\&amp;=&amp;\\frac13e^{3x}cos4x+\\frac43\\int sin4x\\frac{\\operatorname d({\\displaystyle\\frac13}e^{3x})}{\\operatorname dx}dx\\;\\;\\boxed5\\\\&amp;=&amp;\\frac13e^{3x}cos4x+\\frac43{\\frac13e^{3x}\\sin4x-\\frac43\\int e^3xcos4xdx}\\;\\boxed5\\\\&amp;=&amp;\\frac13e^{3x}cos4x+\\frac49e^3xsin4x-\\frac{16}9\\int e^3xcos4xdx\\\\\\int e^3xcos4xdx&amp;=&amp;e^3x(3cos4x+4sin4x)\\;\\boxed5\\\\\\int e^3xcos4xdx&amp;=&amp;\\frac{e^{3x}}{25}(3cos4x+4sin4x)+C\\;\\;\\;\\;\\;\\boxed5\\end{array}<\/span>\n\n\n\n<p>2. \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dba\u0ddc\u0daf\u0d9c\u0db1\u0dd2\u0db8\u0dd2\u0db1\u0dca&nbsp;&nbsp; \u0283 e<sup>4x<\/sup>sin3x dx \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1.&nbsp;&nbsp;&nbsp;&nbsp; (2006)<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int e^4xsin3xdx&amp;=&amp;\\int sin3x\\frac{\\operatorname d{({\\displaystyle\\frac14}e^{4x})}}{\\operatorname dx}dx\\;\\boxed5\\\\&amp;=&amp;{\\frac14e^{4x}}sin3x-\\frac34\\int cos3xe^4xdx\\;\\;\\;\\;\\boxed5\\\\&amp;=&amp;\\frac14e^{4x}sin3x-\\frac34\\int cos3x\\frac{\\operatorname d{({\\displaystyle\\frac14}e^{4x})}}{\\operatorname dx}dx\\;\\;\\;\\;\\boxed5\\\\&amp;=&amp;\\frac14e^{4x}sin3x-\\frac34\\{{\\frac14\\;e^{4x}\\cos3x-\\frac34\\int e^4x(-\\sin3x)dx\\;\\;}\\;\\;\\boxed5\\\\&amp;=&amp;\\frac14e^{4x}sin3x-\\frac3{16}e^{4x}cos3x-\\frac9{16}\\int e^4xsin3xdx\\;\\;\\;\\\\\\int e^4xsin3xdx&amp;=&amp;e^4x(4sin3x-3cos3x)\\;\\;\\;\\;\\boxed5\\\\\\int e^4xsin3xdx&amp;=&amp;\\frac{e^{4x}}{25}(4sin3x\u20133cos3x)+C\\;\\;\\;\\;\\boxed5\\end{array}<\/span>\n\n\n\n<p>3. I = \u0283 e<sup>ax<\/sup> cosbx dx&nbsp;&nbsp; \u0dc4\u0dcf&nbsp; J = \u0283 e<sup>ax<\/sup> sinbx dx \u0dba\u0dd0\u0dba\u0dd2 \u0d9c\u0db1\u0dd2\u0db8\u0dd4. \u0db8\u0dd9\u0dc4\u0dd2 a \u0dc4\u0dcf b \u0dba\u0db1\u0dd4 \u0dc1\u0dd4\u0db1\u0dca\u0dba \u0db1\u0ddc\u0dc0\u0db1 \u0dad\u0dcf\u0dad\u0dca\u0dc0\u0dd2\u0d9a \u0dc3\u0d82\u0d9b\u0dca\u200d\u0dba\u0dcf \u0dc0\u0dda.<\/p>\n\n\n\n<p>bI + aJ = e<sup>ax<\/sup> sinbx&nbsp; \u0dc4\u0dcf&nbsp; aI \u2013 bJ = e<sup>ax<\/sup> cosbx dx &nbsp;\u0db6\u0dc0 \u0db4\u0dd9\u0db1\u0dca\u0dc0\u0db1\u0dca\u0db1. \u0d92\u0db1\u0dba\u0dd2\u0db1\u0dca I&nbsp; \u0dc4\u0dcf&nbsp; J&nbsp; \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1. &nbsp;(2010)<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}I&amp;=&amp;\\int e^{ax}cosbxdx\\\\I&amp;=&amp;\\int cosbx\\frac{\\operatorname d{({\\displaystyle\\frac1a}e^{ax})}}{\\operatorname dx}dx\\;\\;\\;\\;\\boxed5\\\\I&amp;=&amp;\\frac1ae^{ax}cosbx-\\int b(-sinbx)\\bullet\\frac1ae^{ax}dx\\\\I&amp;=&amp;\\frac1ae^{ax}cosbx+\\frac ba\\int sinbxe^{ax}dx\\;\\;\\;\\;\\;\\boxed5\\;\\end{array}<\/span>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}aI-bJ&amp;=&amp;e^{ax}cosbx\\\\J&amp;=&amp;\\int e^{ax}sinbxdxJ=\\int sinbx\\frac{\\operatorname d{({\\displaystyle\\frac1a}e^{ax})}}{\\operatorname dx}dx\\;\\;\\;\\;\\boxed5\\\\J&amp;=&amp;\\frac1ae^{ax}\\sin bx-\\int bcosbx\\bullet\\frac1ae^{ax}dx\\\\&amp;=&amp;\\frac1ae^{ax}\\sin bx-\\frac ba\\int e^axcosbxdx\\;\\;\\;\\;\\;\\;\\;\\boxed5\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\n\n\n\n<p>bI + aJ = e<sup>ax <\/sup>sinbx<\/p>\n\n\n\n<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}aI-bJ&amp;=&amp;cosbx\\bullet e^{ax\\;\\;}\\longrightarrow\\boxed1\\\\aJ+bI&amp;=&amp;sinbx\\bullet e^{ax\\;}\\;\\;\\longrightarrow\\boxed2\\\\&amp;&amp;(1)\\times a+(2)\\times b\\\\(a^2+b^2)I&amp;=&amp;e^{ax}\\lbrack acosbx+bsinbx\\rbrack\\\\I&amp;=&amp;\\frac{e^{ax}}{(a^2+b^2)}\\lbrack acosbx+bsinbx\\rbrack\\;\\;\\boxed5\\\\&amp;&amp;(2)\\times a-(1)\\times b\\\\(a^2+b^2)J&amp;=&amp;e^{ax}\\lbrack asinbx-bcosbx\\rbrack\\\\J&amp;=&amp;\\frac{e^{ax}}{(a^2+b^2)}\\lbrack asinbx\u2013bcosbx\\rbrack\\;\\;\\;\\boxed5\\\\&amp;&amp;\\\\&amp;&amp;\\end{array}<\/span>\r\n<div class=\"epyt-video-wrapper\"><iframe loading=\"lazy\"  id=\"_ytid_60727\"  width=\"696\" height=\"392\"  data-origwidth=\"696\" data-origheight=\"392\"  data-relstop=\"1\" src=\"https:\/\/www.youtube.com\/embed\/UUMj3H5UsKg?enablejsapi=1&autoplay=0&cc_load_policy=0&cc_lang_pref=&iv_load_policy=1&loop=0&rel=0&fs=1&playsinline=0&autohide=2&theme=dark&color=red&controls=1&\" class=\"__youtube_prefs__  no-lazyload\" title=\"YouTube player\"  allow=\"fullscreen; accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen data-no-lazy=\"1\" data-skipgform_ajax_framebjll=\"\"><\/iframe><\/div>\n\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\">\u201cScience is the Differential Calculus of the mind. \u00a0Art the Integral Calculus; \u00a0they may be beautiful when apart, \u00a0but are greatest only when combined.\u201d<\/span><\/strong><\/em><\/span><br \/><span style=\"font-family: tahoma, arial, helvetica, sans-serif;font-size: 10pt;color: #808080\">-Ronald Ross-<\/span><\/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>\u0d9c\u0dd0\u0da7\u0dbd\u0dd4 \u0dc0\u0dd2\u0dc3\u0db3\u0dd3\u0db8 \u0dc3\u0db3\u0dc4\u0dcf \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba \u0db8\u0dd9\u0db8 \u0db4\u0dcf\u0da9\u0db8\u0dda\u0daf\u0dd3  \u0d85\u0db0\u0dca\u200d\u0dba\u0db1\u0dca\u200d\u0dba\u0dba \u0d9a\u0dbb\u0dba\u0dd2.<\/p>\n","protected":false},"author":21,"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":[2503,1288,63,3701,3721,3761,4054,4060,4055,3792,3795,4052,4053,4058,4059,3703,61,3702,4056,3796],"class_list":{"0":"post-12030","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-advace-level","13":"tag-advance-level","14":"tag-al","15":"tag-anukalanaya","16":"tag-com-maths","17":"tag-combined-maths","18":"tag-integration","19":"tag-kalamaya","20":"tag-kotaswashayen-anukalanaya","21":"tag-pure","22":"tag-pure-maths","23":"tag-sanyuktha-ganithaya","24":"tag-shudda-gamithaya","25":"tag-usapela","26":"tag-usaspela-ganithaya","27":"tag-3703","28":"tag-61","29":"tag-3702","30":"tag-4056","31":"tag-3796"},"_links":{"self":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12030"}],"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\/21"}],"replies":[{"embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/comments?post=12030"}],"version-history":[{"count":68,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12030\/revisions"}],"predecessor-version":[{"id":34724,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12030\/revisions\/34724"}],"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=12030"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/categories?post=12030"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/tags?post=12030"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}