{"id":3361,"date":"2024-01-30T12:09:45","date_gmt":"2024-01-30T06:09:45","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=3361"},"modified":"2024-11-07T15:38:19","modified_gmt":"2024-11-07T09:38:19","slug":"integration","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/integration\/","title":{"rendered":"\u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3, \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995, \u0985\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7\u09b0 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u0995\u09cc\u09b6\u09b2, \u09af\u09cb\u0997\u099c\u09c7\u09b0 \u09af\u09cb\u0997\u09b6\u09cd\u09b0\u09df\u09c0 \u09a7\u09b0\u09cd\u09ae"},"content":{"rendered":"<h2><span style=\"color: #339966;\"><b>\u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 <\/b><b>(Integration)<\/b><\/span><\/h2>\n<h3><span style=\"color: #800080;\"><b>\u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u0995 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09af\u09cb\u0997\u099c <\/b><b>(Anti-derivative as Integral)<\/b><b>:<\/b><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">F (<\/span><span style=\"font-weight: 400;\">x)<\/span><span style=\"font-weight: 400;\">\u00a0\u098f\u0995\u099f\u09bf \u0985\u09a8\u09cd\u09a4\u09b0\u099c <\/span><span style=\"font-weight: 400;\">F'(x) = f(<\/span><span style=\"font-weight: 400;\">x)<\/span><span style=\"font-weight: 400;\"> \u0985\u09b0\u09cd\u09a5\u09be\u09ce <span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\{F(x)\\}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b9\u09b2\u09c7, <\/span><span style=\"font-weight: 400;\">F(x)<\/span><span style=\"font-weight: 400;\"> \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf\u0995\u09c7 <\/span><span style=\"font-weight: 400;\">f(x)<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u099c \u0985\u09a5\u09ac\u09be \u0985\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c (<\/span><span style=\"font-weight: 400;\">Indefinite\u00a0Integral<\/span><span style=\"font-weight: 400;\">) \u09ac\u09be \u09b8\u09ae\u09be\u0995\u09b2\u09bf\u09a4 \u09ae\u09be\u09a8 \u09ac\u09b2\u09be \u09b9\u09df \u098f\u09ac\u0982 \u0987\u09b9\u09be\u0995\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">\\int f(x) d x=F(x)<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09b8\u0982\u0995\u09c7\u09a4 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09be \u09b9\u09df\u0964 \u0995\u09cb\u09a8\u09cb \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09af\u09cb\u0997\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09be\u09b0 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u0995\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 (<\/span><span style=\"font-weight: 400;\">Integration<\/span><span style=\"font-weight: 400;\">) \u09ac\u09b2\u09be \u09b9\u09df\u0964 \u098f\u0996\u09be\u09a8\u09c7,<span class=\"katex-eq\" data-katex-display=\"false\">\\int<\/span> <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u09aa\u09cd\u09b0\u09a4\u09c0\u0995 \u098f\u0995\u099f\u09bf \u09b2\u09ae\u09cd\u09ac\u09be <\/span><span style=\"font-weight: 400;\">S<\/span><span style=\"font-weight: 400;\"> \u09ac\u09c1\u099d\u09be\u09df, \u09af\u09be\u09b9\u09be \u2018<\/span><span style=\"font-weight: 400;\">Summation<\/span><span style=\"font-weight: 400;\">\u2019 \u09b6\u09ac\u09cd\u09a6\u099f\u09bf\u09b0 \u09aa\u09cd\u09b0\u09a5\u09ae \u0985\u0995\u09cd\u09b7\u09b0, \u09ab\u09be\u0982\u09b6\u09a8 <\/span><span style=\"font-weight: 400;\">f(x)<\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 \u09af\u09cb\u099c\u09cd\u09af \u09b0\u09be\u09b6\u09bf (<\/span><span style=\"font-weight: 400;\">Integrand<\/span><span style=\"font-weight: 400;\">) \u09ac\u09b2\u09c7 \u098f\u09ac\u0982 \u0985\u09a8\u09cd\u09a4\u09b0\u0995 <\/span><span style=\"font-weight: 400;\">dx<\/span><span style=\"font-weight: 400;\"> \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7 \u09af\u09c7, <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/academic\/10\/\"> \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3<\/a><\/span> \u099a\u09b2\u0995\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}<\/span><\/span><span style=\"font-weight: 400;\"> \u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">\\int d x<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09df\u09be \u09af\u09be\u09b0\u09be \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0 \u09aa\u09b0\u09b8\u09cd\u09aa\u09b0\u0995\u09c7 \u09aa\u09cd\u09b0\u09b6\u09ae\u09bf\u09a4 \u0995\u09b0\u09c7\u0964<\/span><\/p>\n<p><b>\u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 <\/b><b>(Constant of Integration):<\/b><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\{F(x)\\}=f(x) \\text { \u09b9\u09b2\u09c7, } \\frac{d}{d x}\\{F(x)+c\\}=\\frac{d}{d x}\\{F(x)\\}+0 \\Rightarrow \\frac{d}{d x}\\{F(x)+c\\}=f(x)<\/span>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Rightarrow F(x)+c=\\int f(x) d x<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\int f(x) d x=F(x)+c<\/span>,\u00a0 <\/span><span style=\"font-weight: 400;\">\u09af\u09c7\u0996\u09be\u09a8\u09c7, \u2018<\/span><span style=\"font-weight: 400;\">c<\/span><span style=\"font-weight: 400;\">\u2019 \u098f\u0995\u099f\u09bf \u0987\u099a\u09cd\u099b\u09be\u09a7\u09c0\u09a8 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09af\u09be\u0995\u09c7 <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=oWK3OUXDWK4&amp;list=PL0dr4HGr8HPjFX3BK1u_q0RXQ9LpYsjuF\" target=\"_blank\" rel=\"noopener\">\u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3<\/a><\/span> \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u09ac\u09b2\u09be \u09b9\u09df\u0964 \u0985\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c\u09c7\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u09a7\u09cd\u09b0\u09c1\u09ac\u0995 \u0985\u09ac\u09b6\u09cd\u09af\u0987 \u09b2\u09bf\u0996\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/span><\/p>\n<p><b>\u09b2\u0995\u09cd\u09b7\u09a3\u09c0\u09df:<\/b><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}=D \\text { \u09b9\u09b2\u09c7, } D\\{F(x)\\}=f(x) \\Rightarrow D^{-1}\\{f(x)\\}=F(x)+c<\/span><\/li>\n<\/ul>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\int f(x) d x=F(x)+c<\/span>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left[\\int f(x) d x\\right]=\\frac{d}{d x}[F(x)+c]=\\frac{d}{d x}\\{F(x)\\}+0=f(x)<\/span><\/li>\n<\/ul>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\frac{d}{d x}=\\left[\\int f(x) d x\\right]=f(x<\/span>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\int\\left[\\frac{d}{d x}\\{f(x)\\}\\right] d x=\\int f(x) d x=F(x)+c<\/span><\/li>\n<\/ul>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\int\\left[\\frac{d}{d x}\\{f(x)\\}\\right] d x=F(x)+c<\/span>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">y=f(x) \\therefore \\frac{d y}{d x}=f^{\\prime}(x) \\Rightarrow d y=f^{\\prime}(x) d x \\Rightarrow d\\{f(x)\\}=f^{\\prime}(x) d x[\\because y=f(x)]<\/span><\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">\u0989\u09a6\u09be\u09b9\u09b0\u09a3\u09b8\u09cd\u09ac\u09b0\u09c2\u09aa: <span class=\"katex-eq\" data-katex-display=\"false\">(\\sin x)=\\frac{d}{d x}(\\sin x) d x=\\cos x d x, d\\left(3 x^{2}+2 x+5\\right)=(6 x+2) d x<\/span><\/span><\/p>\n<p><b>\u0985\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7\u09b0 \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u0995\u09cc\u09b6\u09b2 <\/b><b>(Different ways to determine Indefinite Integral)<\/b><b>:<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09df\u09be \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09df\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u098f\u09ac\u0982 \u098f \u09a6\u09c1\u099f\u09bf\u09b0 \u09ae\u09be\u099d\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 \u09a6\u09c7\u0996\u09be\u09a8\u09cb\u09b0 \u099c\u09a8\u09cd\u09af \u09a8\u09bf\u099a\u09c7 \u0995\u09a4\u0997\u09c1\u09b2\u09bf \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09af\u09cb\u0997\u099c \u0993 \u0985\u09a8\u09cd\u09a4\u09b0\u099c \u09a6\u09c1\u099f\u09bf\u0987 \u09a6\u09c7\u0996\u09be\u09a8\u09cb \u09b9\u09b2\u09cb\u0964<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left(x^{n}\\right)=n x^{n-1} \\Rightarrow d\\left(x^{n}\\right)=n x^{n-1} d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\int x^{n} d x=\\frac{x^{n+1}}{n+1}+c,(n \\neq-1)<\/span><\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">\u0985\u09a8\u09c1: <span class=\"katex-eq\" data-katex-display=\"false\">\\text { (a) } \\int \\frac{1}{x^{n}} d x=\\int x^{-n} d x=\\frac{x^{-n+1}}{-n+1}+c=\\frac{1}{(1-n) x^{n-1}}+c<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a8\u09c1: <span class=\"katex-eq\" data-katex-display=\"false\">\\text { (b) } \\int d x=\\int x^{n} d x=\\frac{x^{0+1}}{0+1}+c=x+c<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\int d(\\sin x)=\\sin x+c, \\int d\\left(2 x^{3}-3 x\\right)=2 x^{3}-3 x+c<\/span>\n<p><span style=\"font-weight: 400;\">\u0985\u09a8\u09c1: <span class=\"katex-eq\" data-katex-display=\"false\">\\text { (c) } \\int \\frac{1}{\\sqrt{x}} d x=\\int x^{-\\frac{1}{2}} d x=\\frac{x^{-\\frac{1}{2}+1}}{-\\frac{1}{2}+1}+c=\\frac{x^{\\frac{1}{2}}}{\\frac{1}{2}}+c=2 \\sqrt{x}+c<\/span><\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">d\/dx (ln\u2061|x|)=1\/x\u21d2d(ln\u2061|x|)=1\/x dx <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{1}{x} d x=\\ln |x|+c, x \\neq 0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left(e^{m x}\\right)=m e^{m x} \\Rightarrow \\frac{1}{m} d\\left(e^{m x}\\right)=e^{m x} d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int e^{m x} d x=\\frac{1}{m} e^{m x}+c, x \\neq 0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left(a^{x}\\right)=a^{x} \\ln a \\Rightarrow \\frac{1}{\\ln a} d\\left(a^{x}\\right)=a^{x} d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int a^{x} d x=\\frac{a^{x}}{\\ln a}+c, a&gt;0, a \\neq 1<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}(\\sin x)=\\cos x \\Rightarrow d(\\sin x)=\\cos x d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\cos x d x=\\sin x+c<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}(\\cos x)=-\\sin x \\Rightarrow-d(\\cos x)=\\sin x d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\sin x d x=-\\cos x+c<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}(\\tan x)=\\sec ^{2} x \\Rightarrow d(\\tan x)=\\sec ^{2} x d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\sec ^{2} x d x=\\tan x+c<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}(\\cot x)=-\\operatorname{cosec}^{2} x \\Rightarrow-d(\\cot x)=\\operatorname{cosec}^{2} x d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\operatorname{cosec}^{2} x d x=-\\cot x+c<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\cdot \\frac{d}{d x}(\\operatorname{cosec} x)=-\\operatorname{cosec} x \\cot x \\Rightarrow-d(\\operatorname{cosex})=\\operatorname{cosec} x \\cot x d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\operatorname{cosec} x \\cot x d x=-\\operatorname{cosec} x+c<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}(\\sec x)=\\sec x \\tan x \\Rightarrow d(\\sec x)=\\sec x \\tan x d x<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\sec x \\tan x d x=\\sec x+c<\/span><\/li>\n<\/ul>\n<p><b>\u0985\u09a8\u09c1\u09b8\u09bf\u09a6\u09cd\u09a7\u09be\u09a8\u09cd\u09a4:<\/b><b>\u00a0<\/b><\/p>\n<p><i><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\{\\sin (a x+b)\\}=a \\cos (a x+b) \\Rightarrow \\frac{1}{a} d\\{\\sin (a x+b)\\}=\\cos (a x+b) d x<\/span> <\/span><\/i><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\int \\cos (a x+b) d x=\\frac{1}{a} \\sin (a x+b)+c \\text { \u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7, } \\int \\sin a x d x=-\\frac{1}{a} \\cos a x+c,<\/span>\n<p><i><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\int \\sec ^{2} a x d x=\\frac{1}{a} \\tan a x+c, \\int \\operatorname{cosec}^{2} a x d x=-\\frac{1}{a} \\operatorname{cotax}+c<\/span> <\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left(\\sin ^{-1} x\\right)=\\frac{1}{\\sqrt{1-x^{2}}} \\Rightarrow d\\left(\\sin ^{-1} x\\right)=\\frac{1}{\\sqrt{1-x^{2}}} d x \\quad \\int \\frac{1}{\\sqrt{1-x^{2}}} d x=\\sin ^{-1} x+c<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left(\\tan ^{-1} x\\right)=\\frac{1}{1+x^{2}} \\Rightarrow d\\left(\\tan ^{-1} x\\right)=\\frac{1}{1+x^{2}} d x \\quad \\int \\frac{1}{1+x^{2}} d x=\\tan ^{-1} x+c<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left(\\sec ^{-1} x\\right)=\\frac{1}{x \\sqrt{x^{2}-1}} \\Rightarrow d\\left(\\sec ^{-1} x\\right)=\\frac{1}{x \\sqrt{x^{2}-1}} d x \\quad \\int \\frac{1}{x \\sqrt{x^{2}-1}} d x=\\sec ^{-1} x+c<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><b>\u09af\u09cb\u0997\u099c\u09c7\u09b0 \u09af\u09cb\u0997\u09be\u09b6\u09cd\u09b0\u09df\u09c0 \u09a7\u09b0\u09cd\u09ae <\/b><b>(Linear law of Integrals)<\/b><b>:<\/b><\/p>\n<p><b>(i) <\/b><span style=\"font-weight: 400;\">u,\u00a0v,\u00a0w,\u00a0\u2026<\/span><span style=\"font-weight: 400;\"> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf <\/span><span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\"> \u098f\u09b0 \u09ab\u09be\u0982\u09b6\u09a8 \u09b9\u09b2\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">\\int(u+v+w+\\cdots) d x=\\int u d x+\\int v d x+\\int w d x+\\cdots<\/span><\/span><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof):<\/b><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left[\\int u d x+\\int v d x+\\int w d x+\\cdots\\right]=\\frac{d}{d x}\\left[\\int u d x\\right]+\\frac{d}{d x}\\left[\\int v d x\\right]+\\frac{d}{d x}\\left[\\int w d x\\right]+\\cdots<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\Rightarrow \\frac{d}{d x}\\left[\\int u d x+\\int v d x+\\int w d x+\\cdots\\right]=u+v+w+\\cdots<\/span>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a4\u098f\u09ac, \u09af\u09cb\u0997\u099c\u09c7\u09b0 \u09b8\u0982\u099c\u09cd\u099e\u09be \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\">\\int u d x+\\int v d x+\\int w d x+\\cdots=\\int(u+v+w+\\cdots) d x<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\int(u+v+w+\\cdots) d x=\\int u d x+\\int v d x+\\int w d x+\\cdots<\/span>\n<p>&nbsp;<\/p>\n<p><b>(ii) <span class=\"katex-eq\" data-katex-display=\"false\">\\int c f(x) d x=c \\int f(x) d x<\/span><\/b><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof):<\/b><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}\\left[\\int c f(x) d x\\right]=c \\frac{d}{d x}\\left[\\int f(x) d x\\right] \\Rightarrow \\frac{d}{d x}\\left[\\int c f(x) d x\\right]=c f(x)<\/span>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09a4\u098f\u09ac, \u09af\u09cb\u0997\u099c\u09c7\u09b0 \u09b8\u0982\u099c\u09cd\u099e\u09be \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\">\\int c f(x) d x=c \\int f(x) d x<\/span><\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">\u09a8\u09bf\u09df\u09ae-1: \u09af\u09a6\u09bf \u0995\u09cb\u09a8\u09cb \u09af\u09cb\u099c\u09cd\u09af \u09ab\u09be\u0982\u09b6\u09a8\u09c7 <\/span><span style=\"font-weight: 400;\">????<\/span><span style=\"font-weight: 400;\"> \u0985\u09a5\u09ac\u09be <\/span><span style=\"font-weight: 400;\">????<\/span><span style=\"font-weight: 400;\"> \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 \u0998\u09be\u09a4 \u0986\u0995\u09be\u09b0\u09c7 \u0985\u09a5\u09ac\u09be <\/span><span style=\"font-weight: 400;\">????<\/span><span style=\"font-weight: 400;\"> \u0993 <\/span><span style=\"font-weight: 400;\">????<\/span><span style=\"font-weight: 400;\"> \u0997\u09c1\u09a3 \u0986\u0995\u09be\u09b0\u09c7 \u09a5\u09be\u0995\u09c7 \u09a4\u09ac\u09c7 \u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09ac\u09be\u09b0 \u0986\u0997\u09c7\u0987 \u09a4\u09be\u09a6\u09c7\u09b0 \u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u0995\u09cb\u09a3\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u0986\u0995\u09be\u09b0\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df\u0964 \u09af\u09c7\u09ae\u09a8:-<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\sin ^{2} x=\\frac{1}{2}(1-\\cos 2 x) \\quad \\cos ^{2} x=\\frac{1}{2}(1+\\cos 2 x)<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\sin ^{3} x=\\frac{1}{4}(3 \\sin x-\\sin 3 x) \\quad \\cos ^{3} x=\\frac{1}{4}(3 \\cos x+\\cos 3 x)<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\sin A \\cos B=\\frac{1}{2}[\\sin (A+B)+\\sin (A-B)] \\quad \\cos A \\sin B=\\frac{1}{2}[\\sin (A+B)-\\sin (A-B)]<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\cos A \\cos B=\\frac{1}{2}[\\cos (A+B)+\\cos (A-B)] \\quad \\sin A \\sin B=\\frac{1}{2}[\\cos (A-B)-\\cos (A+B)]<\/span>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e7: \u09a8\u09bf\u099a\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u0997\u09c1\u09b2\u09bf\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0:<\/b><\/p>\n<p><b>(Example \u2013 1: Determine the following integrals)<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<span class=\"katex-eq\" data-katex-display=\"false\">\\int x(1+\\sqrt{x}) d x<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{d x}{1+\\cos x}<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(c)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\sin ^{-1} \\frac{1}{2}<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(d)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int 5 \\cos 4 x \\sin 3 x d x<\/span><\/p>\n<p><b>\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int x(1+\\sqrt{x}) d x=\\int\\left(x+x^{\\frac{3}{2}}\\right) d x=\\int x d x+\\int x^{\\frac{3}{2}} d x<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{x^{1+1}}{1+1}+\\frac{x^{\\frac{3}{2}+1}}{\\frac{3}{2}+1}+c<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{x^{2}}{2}+\\frac{x^{\\frac{5}{2}}}{\\frac{5}{2}}+c<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{1}{2} x^{2}+\\frac{2}{5} x^{\\frac{5}{2}}+c<\/span>\u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0<b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{d x}{1+\\cos x}=\\int \\frac{1-\\cos x}{(1+\\cos x)(1-\\cos x)} d x<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int \\frac{1-\\cos x}{1-\\cos ^{2} x} d x=\\int \\frac{1-\\cos x}{\\sin ^{2} x} d x<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int\\left[\\frac{1}{\\sin ^{2} x}-\\frac{\\cos x}{\\sin ^{2} x}\\right] d x=\\int\\left[\\operatorname{cosex}^{2}-\\operatorname{cosec} x \\cdot \\cot x\\right] d x<\/span>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=-\\cot x-(-\\operatorname{cosec} x)+c<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">=\\operatorname{cosec} x-\\cot x+c<\/span>\u00a0 \u00a0 <b>(Ans)<\/b><\/p>\n<p><b>\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Alternative solutions): <\/b><\/p>\n<p><b><span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{d x}{1+\\cos x}=\\int \\frac{d x}{2 \\cos ^{2} \\frac{x}{2}}<\/span>\u00a0<\/b><\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{1}{2} \\int \\sec ^{2} \\frac{x}{2} d x=\\frac{1}{2} \\frac{\\tan \\frac{x}{2}}{\\frac{1}{2}}+c=\\tan \\frac{x}{2}+c<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\text { (c) } \\int \\sec ^{2} x \\operatorname{cosec}^{2} x d x=\\int \\frac{1}{\\cos ^{2} x \\sin ^{2} x} d x<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int \\frac{\\sin ^{2} x+\\cos ^{2} x}{\\cos ^{2} x \\sin ^{2} x} d x=\\int\\left[\\frac{\\sin ^{2} x}{\\cos ^{2} x \\sin ^{2} x}+\\frac{\\cos ^{2} x}{\\cos ^{2} x \\sin ^{2} x}\\right] d x<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int\\left[\\frac{1}{\\cos ^{2} x}+\\frac{1}{\\sin ^{2} x}\\right] d x<\/span>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=\\int\\left[\\sec ^{2} x+\\operatorname{cosec}^{2} x\\right] d x<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">=\\tan x-\\cot x+c<\/span> <b>(Ans)<\/b><\/p>\n<p><b>\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf-1 (Alternative way &#8211; 1):<\/b><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\int \\sec ^{2} x \\operatorname{cosec}^{2} x d x=\\int \\sec ^{2} x\\left(1+\\cot ^{2} x\\right) d x<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int\\left(\\sec ^{2} x+\\sec ^{2} x \\cot ^{2} x\\right) d x<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int\\left(\\sec ^{2} x+\\frac{1}{\\cos ^{2} x} \\times \\frac{\\cos ^{2} x}{\\sin ^{2} x}\\right) d x<\/span>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=\\int\\left(\\sec ^{2} x+\\operatorname{cosec}^{2} x\\right) d x<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">=\\tan x-\\cot x+c<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<b>(Ans)<\/b><\/p>\n<p><b>\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf-2 (Alternative way &#8211; 2): <\/b><\/p>\n<p><b><span class=\"katex-eq\" data-katex-display=\"false\">\\int \\sec ^{2} x \\operatorname{cosec}^{2} x d x=\\int \\frac{1}{\\cos ^{2} x} \\cdot \\frac{1}{\\sin ^{2} x} d x<\/span><\/b><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int \\frac{4}{(2 \\sin x \\cos x)^{2}} d x<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=4 \\int \\frac{1}{\\sin ^{2} 2 x} d x<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=4 \\int \\operatorname{cosec}^{2} 2 x d x<\/span>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=4\\left(-\\frac{1}{2} \\cot 2 x\\right)+c<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">=-2 \\cot 2 x+c<\/span>\u00a0 \u00a0 \u00a0 \u00a0 <b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">(d)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int 5 \\cos 4 x \\sin 3 x d x=\\int \\frac{5}{2}[\\sin (4 x+3 x)-\\sin (4 x-3 x)] d x<\/span><\/p>\n<p><b><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{5}{2} \\int(\\sin 7 x-\\sin x) d x<\/span><\/b><\/p>\n<p><b><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{5}{2}\\left[-\\frac{\\cos 7 x}{7}-(-\\cos x)\\right]+c<\/span><\/b><\/p>\n<p><b><span class=\"katex-eq\" data-katex-display=\"false\">=\\frac{5}{2}\\left(\\cos x-\\frac{1}{7} \\cos 7 x\\right)+c<\/span><\/b>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u0985\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09df <\/b><b>(Determining Indefinite Integral)<\/b><b>:<\/b><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf (<\/b><b>Method of substitution):<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09cb\u099c\u09bf\u09a4 \u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7 \u09b8\u09ac\u099a\u09c7\u09df\u09c7 \u09aa\u09cd\u09b0\u09df\u09cb\u099c\u09a8\u09c0\u09df \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf \u09b9\u099a\u09cd\u099b\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u0964 \u09aa\u09cd\u09b0\u09a6\u09a4\u09cd\u09a4 \u09af\u09cb\u099c\u09cd\u09af \u09b0\u09be\u09b6\u09bf\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09cd\u09ad\u09c1\u0995\u09cd\u09a4 \u0995\u09cb\u09a8\u09cb \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09c7 \u098f\u0995\u099f\u09bf \u099a\u09b2\u09b0\u09be\u09b6\u09bf \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09c7 \u09af\u09cb\u099c\u09bf\u09a4 \u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7\u09b0 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u0995\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u09ac\u09b2\u09be \u09b9\u09df\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af: <span class=\"katex-eq\" data-katex-display=\"false\">x=g(z) \\text {\u09b9\u09b2\u09c7, } \\int f(x) d x=\\int f\\{g(z)\\} g^{\\prime}(z) d z<\/span><\/b><\/p>\n<p><b>(Theorem: If <span class=\"katex-eq\" data-katex-display=\"false\">x=g(z) \\text {\u09b9\u09b2\u09c7, } \\int f(x) d x=\\int f\\{g(z)\\} g^{\\prime}(z) d z<\/span>)<\/b><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 <\/b><b>(Theorem proof)<\/b><b>:<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">F(x)=\\int f(x) d x \\Rightarrow \\frac{d}{d x}\\{F(x)\\}=f(x)<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">x=g(z) \\text { \u09ac\u09b2\u09c7, } \\frac{d x}{d z}=g^{\\prime}(z) \\ldots \\ldots(2)<\/span>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09a8, <span class=\"katex-eq\" data-katex-display=\"false\">\\text { \u098f\u0925\u0928, } \\frac{d}{d z}\\{F(x)\\}=\\frac{d}{d x}\\{F(x)\\} \\cdot \\frac{d x}{d z}=f(x) \\cdot g^{\\prime}(z)<\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<span style=\"font-weight: 400;\">[<\/span><span style=\"font-weight: 400;\">(1)<\/span><span style=\"font-weight: 400;\"> \u0993 <\/span><span style=\"font-weight: 400;\">(2)<\/span><span style=\"font-weight: 400;\"> \u09b9\u09a4\u09c7]<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d z}\\{F(x)\\}=f\\{g(z)\\} \\cdot g^{\\prime}(z)[\\because x=g(z)]<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">(x)=\\int f\\{g(z)\\} \\cdot g^{\\prime}(z) d z \\Rightarrow \\int f(x) d x=\\int f\\{g(z)\\} \\cdot g^{\\prime}(z) d z<\/span>\n<p>&nbsp;<\/p>\n<p><b>\u09a6\u09cd\u09b0\u09b7\u09cd\u099f\u09ac\u09cd\u09af (Note):<\/b> <span style=\"font-weight: 400;\">(2)<\/span><span style=\"font-weight: 400;\"> \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987, <span class=\"katex-eq\" data-katex-display=\"false\">d x=g^{\\prime}(z) d z \\Rightarrow d\\{g(z)\\}=g^{\\prime}(z) d z<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\int f(x) d x=\\int f\\{g(z)\\} \\cdot g^{\\prime}(z) d z=\\int f\\{g(z)\\} d\\{g(z)\\}<\/span>\n<p>&nbsp;<\/p>\n<p><b>\u09a7\u09b0\u09a8-\u09e7 (Type &#8211; 1): <\/b><span style=\"font-weight: 400;\">\u09af\u09cb\u099c\u09cd\u09af \u09b0\u09be\u09b6\u09bf\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">f(x)=g(x) \\times g^{\\prime}(x) \\text { \u09ac\u09be, }\\{g(x)\\}^{n} \\times g^{\\prime}(x) \\text { \u09ac\u09be, } T\\{g(x)\\} . g^{\\prime}(x)<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u0996\u09be\u09a8\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">T=\\sin , \\cos , \\sec ^{2} \\ldots \\text { \u09ac\u09be, }[T\\{g(x)\\}]^{n} \\times g^{\\prime}(x) \\text {\u09ac\u09be, } e^{T\\{g(x)\\}} \\cdot g^{\\prime}(x)<\/span><\/span><span style=\"font-weight: 400;\"> \u0987\u09a4\u09cd\u09af\u09be\u09a6\u09bf \u0986\u0995\u09be\u09b0\u09c7 \u09a5\u09be\u0995\u09c7\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09af\u09a6\u09bf \u09af\u09cb\u099c\u09cd\u09af \u09b0\u09be\u09b6\u09bf\u00a0 f(x) <\/span><span style=\"font-weight: 400;\">\u098f\u09b0 \u098f\u0995 \u0985\u0982\u09b6 <\/span><span style=\"font-weight: 400;\">g(x)<\/span><span style=\"font-weight: 400;\"> \u0995\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09b2\u09c7 \u0985\u09aa\u09b0 \u0985\u0982\u09b6 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df, \u09a4\u09ac\u09c7 <\/span><span style=\"font-weight: 400;\">g(<\/span><span style=\"font-weight: 400;\">x) <\/span><span style=\"font-weight: 400;\">= z<\/span><span style=\"font-weight: 400;\"> \u09a7\u09b0\u09c7 \u09b0\u09be\u09b6\u09bf\u099f\u09bf\u09b0 \u09af\u09cb\u099c\u09bf\u09a4 \u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09be \u09af\u09be\u09df\u0964<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e8: \u09a8\u09bf\u099a\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u0997\u09c1\u09b2\u09bf\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0:<\/b><\/p>\n<p><b>(Example \u2013 2: Determine the following integrals)<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">&lt;span style=&quot;font-weight: 400;&quot;&gt;\\int(3-2 x)^{4} d x&lt;\/span&gt;<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\cos x e^{\\sin x} d x<\/span><\/p>\n<p><b>\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<\/span> <span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">3-2 x=z \\therefore-2 d x=d z \\Rightarrow d x=-\\frac{1}{2} d z<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234<span class=\"katex-eq\" data-katex-display=\"false\">\\int(3-2 x)^{4} d x=\\int z^{4}\\left(-\\frac{1}{2} d z\\right)<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=-\\frac{1}{2} \\cdot \\frac{z^{4+1}}{4+1}+c<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=-\\frac{1}{10}(3-2 x)^{5}+c<\/span><\/span>\u00a0 \u00a0 <b>(Ans)<\/b><\/p>\n<p><b>\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Alternative solutions): <\/b><\/p>\n<p><b><span class=\"katex-eq\" data-katex-display=\"false\">\\int(3-2 x)^{4} d x<\/span><\/b><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">=-\\frac{1}{2} \\int(3-2 x)^{4}(-2 d x), \\quad[\\because d(3-2 x)=-2 d x]<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=-\\frac{1}{2} \\cdot \\frac{(3-2 x)^{4+1}}{4+1}+c<\/span>\n<p>&nbsp;<\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=-\\frac{1}{10}(3-2 x)^{5}+c<\/span>\u00a0 \u00a0 \u00a0 <b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">\\sin x=z \\therefore \\cos x d x=d z<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\int \\cos x e^{\\sin x} d x=\\int e^{\\sin x} \\cos x d x<\/span>\n<p>&nbsp;<\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=\\int e^{z} d z=e^{z}+c=e^{\\sin x}+c \\quad[\\because d(\\sin x)=\\cos x d x]<\/span>\u00a0 \u00a0 \u00a0<b>(Ans)<\/b><\/p>\n<p><b>\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Alternative solutions):<\/b><\/p>\n<p><b><span class=\"katex-eq\" data-katex-display=\"false\">\\int \\cos x e^{\\sin x} d x<\/span><\/b><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int e^{\\sin x}(\\cos x d x)<\/span>\n<p>&nbsp;<\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=e^{\\sin x}+c \\quad[\\because d(\\sin x)=\\cos x d x]<\/span> <b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u09a7\u09b0\u09a8-\u09e8 (Type &#8211; 2):<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{f^{\\prime}(x)}{f(x)} d x=\\int \\frac{d\\{f(x)\\}}{f(x)}=\\ln |f(x)|+c<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{f^{\\prime}(x)}{\\sqrt{f(x)}} d x=\\int \\frac{d\\{f(x)\\}}{\\sqrt{f(x)}}=2 \\sqrt{f(x)}+c<\/span><\/p>\n<p><b>\u09a8\u09bf\u09df\u09ae (Rule):<\/b><b>\u00a0 <\/b><span style=\"font-weight: 400;\">f (<\/span><span style=\"font-weight: 400;\">x) <\/span><span style=\"font-weight: 400;\">= z<\/span><span style=\"font-weight: 400;\"> \u09a7\u09b0\u09a4\u09c7 \u09b9\u09df\u0964<\/span><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 (Proof):<\/b><\/p>\n<p><span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">f(x)=z \\quad \\therefore f^{\\prime}(x) d x=d z<\/span>\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\int \\frac{f^{\\prime}(x)}{f(x)} d x=\\int \\frac{d z}{z}=\\ln |z|+c<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{f^{\\prime}(x)}{\\sqrt{f(x)}} d x=\\int \\frac{d z}{\\sqrt{z}}=2 \\sqrt{z}+c=2 \\sqrt{f(x)}+c<\/span><\/span><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09e9: \u09a8\u09bf\u099a\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u0997\u09c1\u09b2\u09bf\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n<p><b>(Example \u2013 3: Determine the following integrals)<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\tan x d x<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\sec x d x<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(c)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\operatorname{cosec} x d x<\/span><\/p>\n<p><b>\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<\/span> <span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\int \\tan x d x=\\int \\frac{-(-\\sin x d x)}{\\cos x}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=-ln\u2061|cosx|+c<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\ln |\\sec x|+c<\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0<b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\sec x d x<\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{\\sec x(\\sec x+\\tan x)}{(\\sec x+\\tan x)} d x<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{\\sec x \\tan x+\\sec ^{2} x}{(\\sec x+\\tan x)} d x<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\ln |\\sec x+\\tan x|+c<\/span>\u00a0 \u00a0 \u00a0<\/span>\u00a0 \u00a0 <b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">(c)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\operatorname{cosec} x d x<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int \\frac{d x}{\\sin x}=\\int \\frac{d x}{2 \\sin \\frac{x}{2} \\cos \\frac{x}{2}}<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int \\frac{\\sec ^{2} \\frac{x}{2} d x}{\\sec ^{2} \\frac{x}{2} \\cdot 2 \\sin \\frac{x}{2} \\cos \\frac{x}{2}}<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int \\frac{\\left(\\frac{1}{2} \\sec ^{2} \\frac{x}{2}\\right) d x}{\\tan \\frac{x}{2}}<\/span>\n<p>&nbsp;<\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=-\\ln \\left|\\tan \\frac{x}{2}\\right|+c<\/span>\u00a0 \u00a0 \u00a0 <b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a3-\u09ea: \u09a8\u09bf\u099a\u09c7\u09b0 \u09af\u09cb\u0997\u099c\u0997\u09c1\u09b2\u09bf\u09b0 \u09ae\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964<\/b><\/p>\n<p><b>(Example 4 &#8211; Determine the following integrals)<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{1+\\cos x}{x+\\sin x} d x<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{\\tan x}{\\ln \\cos x} d x<\/span><\/p>\n<p><b>\u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Solution):<\/b><\/p>\n<p><span style=\"font-weight: 400;\">(a)<\/span> <span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">x+\\sin x=z \\quad \\therefore(1+\\cos x) d x=d z<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234<span class=\"katex-eq\" data-katex-display=\"false\">\\therefore \\int \\frac{1+\\cos x}{x+\\sin x} d x=\\int \\frac{d z}{z}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\ln |z|+c<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\ln |x+\\sin x|+c<\/span>\u00a0 \u00a0 \u00a0 <\/span><b>(Ans)<\/b><\/p>\n<p><b>\u09ac\u09bf\u0995\u09b2\u09cd\u09aa \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u09a4\u09c7 \u09b8\u09ae\u09be\u09a7\u09be\u09a8 (Alternative solutions):<\/b><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\int \\frac{1+\\cos x}{x+\\sin x} d x=\\int \\frac{d(x+\\sin x)}{x+\\sin x} d x<\/span>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">= <span class=\"katex-eq\" data-katex-display=\"false\">ln\u2061|x+sinx|+c\u00a0 <\/span>\u00a0 <\/span>\u00a0<b>(Ans)<\/b><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">(b)<\/span> <span style=\"font-weight: 400;\">\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">\\ln \\cos x=z \\quad \\therefore \\frac{-\\sin x}{\\cos x} d x=d z \\Rightarrow \\tan x d x=-d z<\/span><\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">=\\int \\frac{\\tan x}{\\ln \\cos x} d x=-\\int \\frac{d z}{z}<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">=-\\ln z+c<\/span>\n<p>&nbsp;<\/p>\n<p><span class=\"katex-eq\" data-katex-display=\"false\">=-\\ln (\\ln \\cos x)+c<\/span>\u00a0 \u00a0<b>(Ans)<\/b><\/p>\n<div style=\"width: 564px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2020\/08\/Integration-formulas.png\" alt=\"\u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3\" width=\"554\" height=\"776\" \/><p class=\"wp-caption-text\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Integration Formulas<\/p><\/div>\n<hr \/>\n<div class=\"x1tlxs6b x1g8br2z x1gn5b1j x230xth x14ctfv x1okitfd x6ikm8r x10wlt62 x1mzt3pk x1y1aw1k xn6708d xwib8y2 x1ye3gou x1n2onr6 x13faqbe x1vjfegm\" role=\"none\">\n<div class=\"\">\n<div class=\"x9f619 x1n2onr6 x1ja2u2z __fb-light-mode\" role=\"none\">\n<p dir=\"auto\" role=\"none\">\n<p class=\"x6prxxf x1fc57z9 x1yc453h x126k92a xzsf02u\" dir=\"auto\" role=\"none\"><em><strong>\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<ul>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-25-online-batch-2-bangla-english-ict\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a \u09e8.\u09e6 (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-26-online-batch-bangla-english-ict\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2025-online-batch\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2026-online-batch\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/medical-admission-course\/\">\u09ae\u09c7\u09a1\u09bf\u0995\u09c7\u09b2 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-a-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-b-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/buet-ques-solve\/\">\u09ac\u09c1\u09df\u09c7\u099f \u0995\u09cb\u09b6\u09cd\u099a\u09c7\u09a8 \u09b8\u09b2\u09ad \u0995\u09cb\u09b0\u09cd\u09b8<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-a-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-b-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u09b8\u09cd\u0995\u09bf\u09b2 \u09a1\u09c7\u09ad\u09c7\u09b2\u09aa\u09ae\u09c7\u09a8\u09cd\u099f \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/study-abroad-complete-guideline\/\">\u09ac\u09bf\u09a6\u09c7\u09b6\u09c7 \u0989\u099a\u09cd\u099a\u09b6\u09bf\u0995\u09cd\u09b7\u09be: Study Abroad Complete Guideline<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/student-hacks\/\">Student Hacks<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ielts-course\/\">IELTS Course by Munzereen Shahid<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/english-grammar-course\/\">Complete English Grammar Course<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ms-bundle\/\"> Microsoft Office 3 in 1 Bundle<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ghore-boshe-freelancing\/\">\u0998\u09b0\u09c7 \u09ac\u09b8\u09c7 Freelancing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/facebook-marketing\/\">Facebook Marketing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/adobe-4-in-1-bundle\/\">Adobe 4 in 1 Bundle<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><em>\u09e7\u09e6 \u09ae\u09bf\u09a8\u09bf\u099f \u09b8\u09cd\u0995\u09c1\u09b2\u09c7\u09b0 \u0995\u09cd\u09b2\u09be\u09b8\u0997\u09c1\u09b2\u09cb \u0985\u09a8\u09c1\u09b8\u09b0\u09a3 \u0995\u09b0\u09a4\u09c7 \u09ad\u09bf\u099c\u09bf\u099f: <span style=\"color: #993300;\"><strong><a style=\"color: #993300;\" href=\"https:\/\/10minuteschool.com\/?ref=https%3A%2F%2Fblog.10minuteschool.com%2Fwordpress%2F&amp;post_id=78178&amp;blog_category_id=700\">www.10minuteschool.com<\/a><\/strong><\/span><\/em><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u09af\u09cb\u0997\u099c\u09c0\u0995\u09b0\u09a3 (Integration) \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u0995 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09af\u09cb\u0997\u099c (Anti-derivative as Integral): F (x)\u00a0\u098f\u0995\u099f\u09bf \u0985\u09a8\u09cd\u09a4\u09b0\u099c F'(x) = f(x) \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u00a0\u09b9\u09b2\u09c7, F(x) \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf\u0995\u09c7 f(x) \u098f\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u0985\u09a8\u09cd\u09a4\u09b0\u099c \u0985\u09a5\u09ac\u09be \u0985\u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09af\u09cb\u0997\u099c (Indefinite\u00a0Integral) \u09ac\u09be \u09b8\u09ae\u09be\u0995\u09b2\u09bf\u09a4 \u09ae\u09be\u09a8 \u09ac\u09b2\u09be \u09b9\u09df \u098f\u09ac\u0982 \u0987\u09b9\u09be\u0995\u09c7 \u00a0\u09b8\u0982\u0995\u09c7\u09a4 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09be \u09b9\u09df\u0964 \u0995\u09cb\u09a8\u09cb \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09af\u09cb\u0997\u099c<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/integration\/\">Read More<\/a><\/p>\n","protected":false},"author":56,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4265,3037,50,3026],"tags":[2396,2391,2393,2389,2395,2394,2392],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3361"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/56"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=3361"}],"version-history":[{"count":42,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3361\/revisions"}],"predecessor-version":[{"id":16139,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/3361\/revisions\/16139"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=3361"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=3361"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=3361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}