{"id":12300,"date":"2021-03-05T20:21:49","date_gmt":"2021-03-05T14:51:49","guid":{"rendered":"https:\/\/learnsteer.sasnaka.org\/science\/?p=12300"},"modified":"2022-02-23T08:11:40","modified_gmt":"2022-02-23T02:41:40","slug":"04-05-08","status":"publish","type":"post","link":"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/04-05-08\/","title":{"rendered":"04.05.08.\u0db1\u0dd2\u0dc1\u0dca\u0da0\u0dd2\u0dad \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<h2 class=\"wp-block-heading\"><strong>\u0d85\u0dbb\u0dca\u0dae \u0daf\u0dd0\u0d9a\u0dca\u0dc0\u0dd3\u0db8<\/strong><\/h2>\n\n\n\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\frac{\\operatorname d\\varnothing(x)}{\\operatorname dx}=f(x)\\;\\text{\u0db1\u0db8\u0dca}\\;\\int f(x)\\;dx\\;=\\varnothing(x)\\;+\\;C\\;\\text{\u0dc0\u0dda.}\\;\\text{\u0d91\u0dba \u0d85\u0db1\u0dd2\u0dc1\u0dca\u0da0\u0dd2\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0dbd\u0dd9\u0dc3 \u0dc4\u0dd0\u0daf\u0dd2\u0db1\u0dca\u0dc0\u0dda. }\\;<\/span>\n<p style=\"text-align: center\"><br><strong><span class=\"wp-katex-eq\" data-display=\"false\">\\int_a^bf\\left(x\\right)\\operatorname dx=\\left[\\phi\\left(x\\right)\\right]_a^b=\\phi\\left(b\\right)-\\phi\\left(a\\right)<\/span><\/strong><br>\u0dbd\u0dd9\u0dc3 \u0dbd\u0dd2\u0dba\u0db1\u0dd4 \u0dbd\u0dd0\u0db6\u0dda.<\/p><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\u0db8\u0dd9\u0dc0\u0dd0\u0db1\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba\u0d9a\u0dca \u0db1\u0dd2\u0dc1\u0dca\u0da0\u0dd2\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba\u0d9a\u0dca \u0dbd\u0dd9\u0dc3 \u0dc4\u0dd0\u0db3\u0dd2\u0db1\u0dca\u0dc0\u0dda.<\/li><li>\u0db8\u0dd9\u0dc4\u0dd2 a \u0dba\u0da7\u0dad\u0dca \u0dc3\u0dd3\u0db8\u0dcf\u0dc0 \u0dc0\u0db1 \u0d85\u0dad\u0dbb b \u0d8b\u0da9\u0dad\u0dca \u0dc3\u0dd3\u0db8\u0dcf\u0dc0 \u0dc0\u0dda.<\/li><\/ul>\n\n\n\n<p>\u0d8b\u0daf\u0dcf:<\/p>\n\n\n\n<p>1.<span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}\\int_1^2x^2dx\\;&amp;=&amp;\\;\\left[\\frac{x^3}3\\right]_1^2\\\\&amp;=&amp;\\;\\frac13(2^3\\;-\\;1^3)\\\\&amp;=&amp;\\frac73\\end{array}<\/span><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>2.<span class=\"wp-katex-eq\" data-display=\"false\">\\int_1^2\\frac x{\\sqrt{x^2+1}}dx<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\frac d{dx}\\left(\\sqrt{x^2+1}\\right)=\\frac12.\\frac1{\\sqrt{x^2+1}}.2x=\\;\\frac x{\\sqrt{x^2+1}}\\Rightarrow\\int\\frac{\\displaystyle x}{\\displaystyle\\sqrt{x^2+1}}dx=\\sqrt{x^2+1}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\"> \\begin{array}{rcl}\\int_0^\\sqrt3\\frac x{\\sqrt{x^2+1}}\\operatorname dx &amp;=&amp;\\left[\\sqrt{x^2+1}\\right]_0^\\sqrt3\\\\\\;&amp;=&amp;\\sqrt{\\left(\\sqrt3\\right)^2+1}-\\sqrt{0+1}\\\\\\;&amp;=&amp;2-1\\\\\\;&amp;=&amp;1\\end{array} <\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><\/p>\n\n\n\n<p class=\"has-text-align-center\"><\/p>\n\n\n\n\n\n<p><p>3.<span class=\"wp-katex-eq\" data-display=\"false\"> \\begin{array}{rcl}\\int_1^4\\frac{e^\\sqrt x}{\\sqrt x}\\operatorname dx &amp;=&amp;2\\int_1^4\\frac{e^\\sqrt x}{2\\sqrt x}\\operatorname dx \\\\\\;&amp;=&amp;2\\left[e^\\sqrt x\\right]_1^4\\\\\\;&amp;=&amp;2\\left(e^\\sqrt4-e^\\sqrt1\\right)\\\\\\;&amp;=&amp;2\\left(e^2-e\\right)\\end{array} <\/span><\/p>\n<p>4.<span class=\"wp-katex-eq\" data-display=\"false\">\\int_0^ 1\\sqrt{4-x^2}\\;dx<\/span><br>\u0db8\u0dd9\u0dba \u0dc0\u0dd2\u0dc3\u0db3\u0dd3\u0db8\u0da7 \u0db4\u0dd9\u0dbb \u0db4\u0dbb\u0dd2\u0daf\u0dd3 &nbsp;&nbsp;x = 2sin\u03b8 \u0dba\u0ddc\u0daf\u0dcf \u0d9c\u0db1\u0dd3.<br>\u0d85\u0db1\u0dd2\u0dc1\u0dca\u0da0\u0dd2\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba\u0dda\u0daf\u0dd3 \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba \u0d9a\u0dbb \u0db4\u0dc3\u0dd4\u0dc0 \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dd9\u0dbb\u0dda. \u0db1\u0db8\u0dd4\u0dad\u0dca \u0db8\u0dd9\u0dc4\u0dd2\u0daf\u0dd3 \u0d8a\u0da7 \u0d85\u0db8\u0dad\u0dbb\u0dc0 \u0dc0\u0dd2\u0da0\u0dbd\u0dca\u200d\u0dba \u0db8\u0dcf\u0dbb\u0dd4 \u0d9a\u0dd9\u0dbb\u0dd9\u0db1 \u0db1\u0dd2\u0dc3\u0dcf \u0d8b\u0da9\u0dad\u0dca \u0dc4\u0dcf \u0dba\u0da7\u0dad\u0dca \u0dc3\u0dd3\u0db8\u0dcf \u0dc0\u0dd9\u0db1\u0dc3\u0dca \u0dc0\u0dda. \u0d91\u0db8 \u0db1\u0dd2\u0dc3\u0dcf \u0d92\u0dc0\u0dcf\u0daf \u0dc3\u0ddc\u0dba\u0dcf \u0d9c\u0dad \u0dba\u0dd4\u0dad\u0dd4\u0dba\u0dd2.<br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}x\\;=\\;2\\sin\\theta\\\\x\\;=\\;1\\;\\text{\u0dc0\u0dd2\u0da7 }\\;\\frac12\\;=\\;\\sin\\theta\\;\\Rightarrow\\theta\\;=\\;\\frac{\\mathrm\\pi}6\\\\x\\;=\\;0\\;\\mathrm{\u0dc0\u0dd2\u0da7}\\;0\\;=\\;\\sin\\theta\\;\\;\\;\\;\\Rightarrow\\;\\theta\\;=\\;0\\\\\\frac{\\operatorname dx}{\\operatorname d\\theta}\\;=\\sin\\theta\\;\\;\\Rightarrow\\;dx\\;=\\;\\sin\\theta d\\theta\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{l}\\begin{array}{rcl}\\int_0^1\\sqrt{4-x^2}\\;dx\\;&amp;=&amp;\\;\\int_0^\\frac\\pi6(\\sqrt{4-4\\sin^2\\theta})\\sin\\theta\\;d\\theta\\\\&amp;=&amp;\\int_0^\\frac\\pi6(\\sqrt{4-4\\sin^2\\theta})\\sin\\theta\\;d\\theta\\\\&amp;=&amp;\\int_0^\\frac\\pi62\\sqrt{\\cos^2\\theta}\\sin\\theta\\;d\\theta\\\\&amp;=&amp;\\int_0^\\frac\\pi62\\cos\\theta\\sin\\theta\\;d\\theta\\\\&amp;=&amp;\\int_0^\\frac\\pi6\\sin2\\theta\\;d\\theta\\\\&amp;=&amp;-\\frac12\\left[\\cos2\\theta\\right]_0^\\frac\\pi6\\\\&amp;=-&amp;\\frac12\\cos2\\left(\\frac{\\mathrm\\pi}6\\right)\\;-\\;\\cos0\\\\&amp;=&amp;-\\frac12(\\frac12-1)\\\\&amp;=&amp;\\frac14\\end{array}\\\\\\\\\\end{array}<\/span><\/p><\/p>\n\n\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>5.<span class=\"wp-katex-eq\" data-display=\"false\">\\int_0^\\frac\\pi2\\frac{dx}{3\\cos x-4\\sin x+5}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}tan\\frac x2&amp;=&amp;t\\;\\text{\u0d86\u0daf\u0dda\u0dc1\u0dba }x=2tan^{-1}t\\\\\\frac{dx}{dt}&amp;=&amp;\\frac2{1+t^2}\\Longrightarrow dx=\\frac{2dt}{1+t^2}\\\\x&amp;=&amp;0\\text{\u0dc0\u0dd2\u0da7}t=tan0=0\\\\x&amp;=&amp;\\frac\\pi2\\text{\u0dc0\u0dd2\u0da7 }t=tan\\frac\\pi4=1\\\\cosx&amp;=&amp;\\frac{1-\\tan^2{\\displaystyle\\frac x2}}{1+tan^2\\frac x2}=\\frac{1-t^2}{1+t^2}\\\\sinx&amp;=&amp;\\frac{2\\tan{\\displaystyle\\frac x2}}{1+tan^2\\frac x2}=\\frac{2t}{1+t^2}\\\\&amp;&amp;\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}&amp;&amp;\\\\\\int_0^\\frac\\pi2\\frac{dx}{3\\cos x-4\\sin x+5}&amp;=&amp;\\int_0^1\\frac{\\displaystyle\\frac{2dt}{1+t^2}}{3(\\frac{1-t^2}{1+t^2})-4(\\frac{2t}{1+t^2})+5}\\\\&amp;=&amp;\\int_0^1\\frac{\\displaystyle\\frac{2dt}{1+t^2}}{3(\\frac{1-t^2}{1+t^2})-4(\\frac{2t}{1+t^2})+5}\\\\&amp;=&amp;\\int_0^1\\frac{2dt}{3(1-t^2)-4(2t)+5(1+t^2)}\\\\&amp;=&amp;\\int_0^1\\frac{dt}{t^2-4t+4}\\\\&amp;=&amp;\\int_0^1\\frac{dt}{(t-2)^2}\\\\&amp;=&amp;\\lbrack-(t-2)^{-1}\\rbrack_0^1\\\\&amp;=&amp;-\\frac1{(-1)}-(\\frac{-1}{-2})\\\\&amp;=&amp;\\frac12\\\\&amp;&amp;\\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<div class=\"wp-block-cover has-pale-pink-background-color has-background-dim\" style=\"min-height:441px;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:#681d1d;font-size:20px\"><strong>\u0daf\u0dd2\u0d9c \u0db8\u0dd3\u0da7\u0dbb\u0dca 1 \u0d9a\u0dca \u0db6\u0dd0\u0d9c\u0dd2\u0db1\u0dca \u0dc0\u0dd6 \u0daf\u0dac\u0dd4 \u0d9a\u0dd0\u0db6\u0dbd\u0dd2 1000 \u0d9a\u0dca \u0dad\u0dd2\u0db6\u0dd9\u0db1\u0dc0\u0dcf. \u0d91\u0d9a\u0dca \u0db4\u0dd4\u0daf\u0dca\u0d9c\u0dbd\u0dba\u0dd9\u0d9a\u0dca \u0db8\u0dd9\u0db8 \u0daf\u0dac\u0dd4 \u0d91\u0d9a\u0dd2\u0db1\u0dca \u0d91\u0d9a\u0dca \u0d9c\u0dd9\u0db1 \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0daf\u0dd9\u0d9a\u0d9a\u0da7 \u0d9a\u0da9\u0db1\u0dc0\u0dcf. \u0dc3\u0dd1\u0db8 \u0daf\u0dab\u0dca\u0da9\u0d9a\u0dca\u0db8 \u0dc0\u0dd9\u0db1\u0dca \u0dc0\u0db1\u0dca\u0db1\u0dda \u0d91\u0d9a\u0dd2\u0db1\u0dd9\u0d9a\u0da7 \u0dc0\u0dd9\u0db1\u0dc3\u0dca \u0daf\u0dd2\u0d9c\u0dc0\u0dbd\u0dd2\u0db1\u0dd4\u0dba\u0dd2.<\/strong><\/p>\n\n\n\n<p class=\"has-text-color\" style=\"color:#681d1d;font-size:20px\"><strong>\u0d89\u0dc4\u0dad \u0dc3\u0dd2\u0daf\u0dca\u0db0\u0dd2\u0dba \u0dc3\u0db8\u0dca\u0db6\u0db1\u0dca\u0db0\u0dc0 \u0d94\u0db6\u0da7 \u0dbd\u0dd0\u0db6\u0dd9\u0db1 \u0d9c\u0dd0\u0da7\u0dc5\u0dd4 \u0db8\u0dda\u0dc0\u0dcf\u0dba\u0dd2.<\/strong><br><strong>1.<\/strong><span class=\"tadv-color\" style=\"color: #681d1d\"><strong>\u0daf\u0dd2\u0d9c\u0dd2\u0db1\u0dca \u0d85\u0da9\u0dd4(\u0daf\u0dd2\u0d9c \u0db8\u0dd3\u0da7\u0dbb\u0dca 0.5 \u0da7 \u0dc0\u0da9\u0dcf \u0d85\u0da9\u0dd4) \u0daf\u0dac\u0dd4 \u0d9a\u0ddc\u0da7\u0dc3\u0dca\u0dc0\u0dbd \u0dc3\u0dcf\u0db8\u0dcf\u0db1\u0dca\u200d\u0dba \u0daf\u0dd2\u0d9c \u0d9a\u0dd3\u0dba\u0daf?<\/strong><br><strong>2.\u0daf\u0dd2\u0d9c\u0dd2\u0db1\u0dca \u0dc0\u0dd0\u0da9\u0dd2 \u0daf\u0dac\u0dd4 \u0d9a\u0ddc\u0da7\u0dc3\u0dca\u0dc0\u0dbd \u0dc3\u0dcf\u0db8\u0dcf\u0db1\u0dca\u200d\u0dba \u0daf\u0dd2\u0d9c \u0d9a\u0dd3\u0dba\u0daf?<\/strong><\/span><br><strong>3.\u0daf\u0dd2\u0d9c\u0dd2\u0db1\u0dca \u0d85\u0da9\u0dd4 \u0daf\u0dac\u0dd4 \u0d9a\u0ddc\u0da7\u0dc3\u0dca\u0dc0\u0dbd \u0dc3\u0dcf\u0db8\u0dcf\u0db1\u0dca\u200d\u0dba\u0dba \u0dc4\u0dcf \u0daf\u0dd2\u0d9c\u0dd2\u0db1\u0dca \u0dc0\u0dd0\u0da9\u0dd2 \u0daf\u0dac\u0dd4 \u0d9a\u0ddc\u0da7\u0dc3\u0dca\u0dc0\u0dbd \u0dc3\u0dcf\u0db8\u0dcf\u0db1\u0dca\u200d\u0dba\u0dba \u0d85\u0dad\u0dbb \u0d85\u0db1\u0dd4\u0db4\u0dcf\u0dad\u0dba \u0d9a\u0dd3\u0dba\u0daf?<\/strong><\/p>\n\n\n\n<p class=\"has-text-color\" style=\"color:#681d1d;font-size:20px\"><strong>\u0d89\u0d9f\u0dd2\u0dba :- 1 \u0dc4\u0dcf 2 \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0db4\u0dc4\u0dc3\u0dd4\u0dc0\u0dd9\u0db1\u0dca \u0dc0\u0dd2\u0dc3\u0db3\u0db1\u0dca\u0db1 \u0db4\u0dd4\u0dc5\u0dd4\u0dc0\u0db1\u0dca. 3 \u0d9a\u0ddc\u0da7\u0dc3\u0dda \u0db4\u0dd2\u0dc5\u0dd2\u0dad\u0dd4\u0dbb \u0dbd\u0db6\u0dcf \u0d9c\u0dd0\u0db1\u0dd3\u0db8\u0da7 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad \u0d9a\u0dc5 \u0dba\u0dd4\u0dad\u0dd4 \u0dc0\u0dd9\u0db1\u0dc0\u0dcf.<\/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 has-custom-font-size is-style-shadow\" style=\"font-size:20px\"><a class=\"wp-block-button__link has-background\" href=\"https:\/\/learnsteer.sasnaka.org\/science\/?p=35695\" style=\"border-radius:10px;background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(104,29,29) 1%)\" 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<\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><span style=\"color: #304170\"><strong>\u0db1\u0dd2\u0dc1\u0dca\u0da0\u0dd2\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba\u0dda \u0dbd\u0d9a\u0dca\u0dc2\u0dab<\/strong><\/span><\/h4>\n\n\n\n<p><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{c}1.\\int_a^bf(x)dx=\\int_a^bf(t)dt\\\\x=t\\\\dx=dt\\\\x\\rightarrow a\\;\\text{\u0dc0\u0dd2\u0da7 }t\\rightarrow a\\\\x\\rightarrow b\\;\\text{\u0dc0\u0dd2\u0da7 }t\\rightarrow b\\\\\\int_a^bf(x)dx=\\int_a^bf(t)dt\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{c}2.\\int_a^bf(x)dx=-\\int_b^af(x)dx\\\\{\\text{\u0dc3\u0dcf\u0db0\u0db1\u0dba}:-f(x)\\text{\u0dc4 \u0dd2\u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0dca\u200d\u0dba }F(x)\\text{\u0dba\u0dd0\u0dba\u0dd2 \u0d9c\u0db1\u0dd2\u0db8\u0dd4}.}\\\\\\int_a^bf(x)dx=\\lbrack F(x)\\rbrack_a^b\\;\\;\\\\=F(b)-F(a)\\\\=-\\lbrack F(a)-F(b)\\rbrack\\\\=-\\lbrack F(x)\\rbrack_b^a\\;\\;\\;\\;\\\\=-\\int_b^af(x)dx\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}3.\\int_a^bf(x)dx&amp;=&amp;\\int_a^cf(x)dx+\\int_c^bf(x)dx\\\\\\text{\u0dc3\u0dcf\u0db0\u0db1\u0dba}:\\int_a^bf(x)dx&amp;=&amp;\\lbrack F(x){\\rbrack_a^b\\;}\\\\&amp;=&amp;F(b)-F(a)\\\\&amp;=&amp;F(b)-F(c)+F(c)-F(a)\\\\&amp;=&amp;\\lbrack F(x)\\rbrack_c^b+\\lbrack F(x)\\rbrack_a^c\\\\&amp;=&amp;\\int_c^bf(x)dx+\\int_a^cf(x)dx\\\\&amp;=&amp;\\int_a^cf(x)dx+\\int_c^bf(x)dx\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}4.\\int_0^af(x)dx&amp;=&amp;\\int_0^af(a-x)dx\\\\\\text{\u0d86\u0daf\u0dda\u0dc1\u0dba }\\\\x&amp;=&amp;a-t\\;\\\\dx&amp;=&amp;-dt\\;\\\\x&amp;\\rightarrow&amp;0\\;\\text{\u0dc0\u0dd2\u0da7 }t\\rightarrow a\\\\x&amp;\\rightarrow&amp;a\\;\\text{\u0dc0\u0dd2\u0da7 }t\\rightarrow0\\\\\\int_0^af(x)dx&amp;=&amp;-\\int_a^0f(a-t)dt\\\\&amp;=&amp;\\int_0^af(a-t)dt=\\int_0^af(a-x)dx;\\text{\u0db4\u0dc5\u0db8\u0dd4 \u0dbd\u0d9a\u0dca\u0dc2\u0dab\u0dba\u0da7 \u0d85\u0db1\u0dd4\u0dc0 }\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}5.\\int_a^bf(x)dx&amp;=&amp;\\int_a^bf(a+b-x)dx\\\\\\text{\u0d86\u0daf\u0dda\u0dc1\u0dba }\\\\x&amp;=&amp;a+b-t\\\\dx&amp;=&amp;-dt\\\\x&amp;\\rightarrow&amp;a\\;\\text{\u0dc0\u0dd2\u0da7 }t\\rightarrow b\\\\x&amp;\\rightarrow&amp;b\\;\\text{\u0dc0\u0dd2\u0da7 }t\\rightarrow a\\\\\\int_a^bf(x)dx&amp;=&amp;-\\int_b^af(a+b-t)dt\\\\&amp;=&amp;\\int_a^bf(a+b-t)dt;\\;\\;\\;\\;\\text{\u0db4\u0dc5\u0db8\u0dd4 \u0dbd\u0d9a\u0dca\u0dc2\u0dab\u0dba\u0da7 \u0d85\u0db1\u0dd4\u0dc0}\\\\&amp;=&amp;\\int_a^bf(a+b-x)dx\\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><\/p>\n\n\n\n<p>\u0d8b\u0daf\u0dcf:-<\/p>\n\n\n\n<p><p>1.(04) \u0dbd\u0d9a\u0dca\u0dc2\u0dab\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca I \u0dc4\u0dd2 \u0d85\u0d9c\u0dba \u0dc3\u0ddc\u0dba\u0db1\u0dca\u0db1<br><span class=\"wp-katex-eq\" data-display=\"false\">\\begin{array}{rcl}I&amp;=&amp;\\int_0^{\\pi\/2}\\frac{Sin^3x}{Cos^3x+Six^3x}dx------(1)\\\\I&amp;=&amp;\\int_0^{\\pi\/2}\\frac{Sin^3({\\displaystyle\\frac\\pi2}-x)dx}{Cos^3(\\frac\\pi2-x)+Sin^3(\\frac\\pi2-x)}\\\\I&amp;=&amp;\\int_0^{\\pi\/2}\\frac{Cos^3x}{Sin^3x+Cos^3x}dx------(2)\\\\&amp;&amp;(1)+(2)\\\\2I&amp;=&amp;\\int_0^{\\pi\/2}\\frac{Sin^3x+Cos^3x}{Sin^3x+Cos^3x}dx\\\\&amp;=&amp;\\int_0^{\\pi\/2}dx\\\\&amp;&amp;\\\\&amp;=&amp;\\lbrack x\\rbrack_0^{\\pi\/2}\\\\&amp;&amp;\\\\&amp;=&amp;\\frac\\pi2-0\\\\&amp;=&amp;\\frac\\pi2\\\\I&amp;=&amp;\\frac\\pi4\\\\&amp;&amp;\\end{array}<\/span><br><span class=\"wp-katex-eq\" data-display=\"false\">2.\\begin{array}{rcl}&amp;&amp;\\int_0^\\pi\\frac{x\\sin xdx}{1+cos^2x}dx\\\\I&amp;=&amp;\\int_0^\\pi\\frac{x\\sin xdx}{1+cos^2x}dx-----(1)\\\\I&amp;=&amp;\\int_0^\\pi\\frac{(\\mathrm\\pi-x)\\sin{(\\mathrm\\pi-x)}dx}{1+cos^2{(\\mathrm\\pi-x)}}\\\\I&amp;=&amp;\\int_0^\\pi\\frac{(\\mathrm\\pi-x)\\sin{(x)}dx}{1+cos^2{(x)}}\\;-----(2)\\\\&amp;&amp;(1)+(2)\\\\2I&amp;=&amp;\\int_0^\\pi\\frac{x\\sin xdx}{1+cos^2x}dx\\;+\\;\\int_0^\\pi\\frac{(\\mathrm\\pi-x)\\sin{(x)}dx}{1+cos^2{(x)}}\\\\2I\\;&amp;=&amp;\\;\\int_0^\\pi\\frac{\\mathrm{\u03c0sin}xdx}{1+cos^2x}dx\\\\I&amp;=&amp;\\frac{\\mathrm\\pi}2\\int_0^\\pi\\frac{\\sin xdx}{1+cos^2x}dx\\\\\\frac{d\\cos x}{dx}\\;&amp;=&amp;\\;-\\sin x\\;\\Rightarrow d\\cos x\\;=\\;-\\sin xdx\\\\\\;I&amp;=&amp;\\frac{\\mathrm\\pi}2\\int_0^\\pi\\frac{-d\\cos x}{1+cos^2x}dx\\\\I&amp;=&amp;-\\frac{\\mathrm\\pi}2\\int_0^\\pi\\frac{d\\cos x}{1+(cosx)^2}dx\\\\&amp;=&amp;-\\frac{\\mathrm\\pi}2\\lbrack tan^{-1}(cos\\pi)\\rbrack_0^{\\mathrm\\pi}\\\\I&amp;=&amp;-\\frac{\\mathrm\\pi}2\\lbrack tan^{-1}(cos\\pi)-tan^{-1}(\\cos0)\\rbrack\\\\I&amp;=&amp;{-\\frac{\\mathrm\\pi}2\\lbrack tan^{-1}(-1)-tan^{-1}(1)\\rbrack\\;}\\\\I&amp;=&amp;-\\frac{\\mathrm\\pi}2\\lbrack\\;\\frac{\\mathrm\\pi}4-(-\\frac{\\mathrm\\pi}4)\\rbrack\\\\I&amp;=&amp;\\frac{\\mathrm\\pi^2}4\\end{array}<\/span><\/p> <\/p>\n\n\n\n<\/p>\r\n<p class=\"has-text-align-center has-background\" style=\"background-color: #272062;text-align: center\"><em><strong><span style=\"font-family: 'book antiqua', palatino, serif;font-size: 18pt;color: #ffffff\">\u00a0&#8220;The theory of probabilities is at the bottom nothing but common sense reduced to calculus&#8221;<\/span><\/strong><\/em> <br \/><span style=\"font-family: tahoma, arial, helvetica, sans-serif;font-size: 10pt;color: #808080\">-Pierre Laplace &#8211;<\/span> <\/p>\r\n<p class=\"has-text-align-center has-background\" style=\"background-color: #272062\">\n\n\n\n<p><div class=\"epyt-video-wrapper\"><iframe loading=\"lazy\"  id=\"_ytid_37392\"  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><\/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>\u0db1\u0dd2\u0dc1\u0dca\u0da0\u0dd2\u0dad \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba\u0dda \u0db8\u0dd6\u0dbd\u0dd2\u0d9a \u0dbd\u0d9a\u0dca\u0dc2\u0dab  \u0dc0\u0dd2\u0db8\u0dc3\u0dd3\u0db8 \u0db8\u0dd9\u0db8 \u0db4\u0dcf\u0da9\u0db8\u0dda \u0d85\u0db1\u0dca\u0dad\u0dbb\u0dca\u0d9c\u0dad \u0dc0\u0dda. <\/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":[2866,4683,63,4684,4109,4682,3701,4054,4060,4681,3792,3794,3795,4236,4091,4090,3947,4059,4685,4686,3703,2608,61,1666,3702,4089,3793,3796,4153],"class_list":{"0":"post-12300","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-advanced-level","13":"tag-advanced-level-maths","14":"tag-al","15":"tag-al-anukalanaya","16":"tag-al-maths","17":"tag-al-pure","18":"tag-anukalanaya","19":"tag-integration","20":"tag-kalamaya","21":"tag-nishchitha-anukalanaya","22":"tag-pure","23":"tag-pure-mathematics","24":"tag-pure-maths","25":"tag-pure-mths","26":"tag-specific-integration-2","27":"tag-specific-integration","28":"tag-usas-pela","29":"tag-usaspela-ganithaya","30":"tag-4685","31":"tag-4686","32":"tag-3703","33":"tag-2608","34":"tag-61","35":"tag-1666","36":"tag-3702","37":"tag-4089","38":"tag-3793","39":"tag-3796","40":"tag-4153"},"_links":{"self":[{"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12300"}],"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=12300"}],"version-history":[{"count":93,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12300\/revisions"}],"predecessor-version":[{"id":35741,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/posts\/12300\/revisions\/35741"}],"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=12300"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/categories?post=12300"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/learnsteer.sasnaka.org\/science\/wp-json\/wp\/v2\/tags?post=12300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}