{"id":9324,"date":"2018-02-09T15:38:33","date_gmt":"2018-02-09T10:08:33","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/?p=9324"},"modified":"2019-03-02T16:25:57","modified_gmt":"2019-03-02T10:55:57","slug":"integrals-class-12-notes-mathematics","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/integrals-class-12-notes-mathematics\/","title":{"rendered":"Integrals Class 12 Notes Mathematics"},"content":{"rendered":"<p><strong>Integrals Class 12 Notes Mathematics<\/strong> in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Integrals class 12 Notes latest chapter wise notes for quick preparation of CBSE board exams and school-based annual examinations. Class 12 Mathematics notes on chapter 7 Integrals are also available for download in CBSE Guide website.<\/p>\n<h2>Integrals Class 12 Notes Mathematics<\/h2>\n<p>Download CBSE class 12th revision notes for chapter 7 Integrals in PDF format for free. Download revision notes for Integrals class 12 Notes and score high in exams. These are the Integrals class 12 Notes prepared by team of expert teachers. The revision notes help you revise the whole chapter 7 in minutes. Revision notes in exam days is one of the best tips recommended by teachers during exam days.<\/p>\n<p style=\"text-align: center;\"><strong><a class=\"button\" href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics\/1284\/cbse-revision-notes\/7\/\">Download Revision Notes as PDF<\/a><\/strong><\/p>\n<h2><strong>CBSE Class 12 Mathematics Revision Notes Chapter 7 Integrals<br \/>\n<\/strong><\/h2>\n<ul>\n<li>Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation. Let.<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image001.png\" \/> Then we write.<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image002.png\" \/> These integrals are called indefinite integrals or general integrals, C is called constant of integration. All these integrals differ by a constant.<\/li>\n<li>From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.<\/li>\n<li>Some properties of indefinite integrals are as follows:<\/li>\n<\/ul>\n<p>1. <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {\\left[ {f\\left( x \\right) + g\\left( x \\right)} \\right]{\\rm{ }}dx} = \\int {f\\left( x \\right){\\rm{ }}dx} + \\int {g\\left( x \\right){\\rm{ }}dx}\" alt=\"Integrals Class 12 Notes Mathematics\" width=\"330\" height=\"37\" \/><\/span><\/span><\/p>\n<p>2. For any real number k, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image004.png\" \/><\/p>\n<p>More generally, if <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image005.png\" \/>\u00a0are functions and <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image006.png\" \/>\u00a0are real numbers. Then<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image007.png\" alt=\"Integrals Class 12 Notes Mathematics\" width=\"265\" height=\"30\" \/><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image008.png\" \/><\/p>\n<ul>\n<li><strong>Some standard integrals:<\/strong><\/li>\n<\/ul>\n<p>(i)\u00a0\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image009.png\" \/>\u00a0<\/strong>Particularly, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image010.png\" \/><\/p>\n<p>(ii)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image011.png\" \/><\/strong><\/p>\n<p>(iii)\u00a0\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image012.png\" \/><\/strong><\/p>\n<p>(iv)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image013.png\" \/><\/strong><\/p>\n<p>(v)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image014.png\" \/><\/strong><\/p>\n<p>(vi)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image015.png\" \/><\/strong><\/p>\n<p>(vii)\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image016.png\" \/><\/strong><\/p>\n<p>(viii)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image017.png\" \/><\/strong><\/p>\n<p>(ix)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image018.png\" \/><\/strong><\/p>\n<p>(x)\u00a0\u00a0\u00a0\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {dx\\over 1+x^2} = tan^{-1}x + C\" \/><\/span><\/span><br \/>\n(xi)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {dx\\over 1+x^2} = -cot^{-1}x + C\" \/><\/span><\/span><\/p>\n<p>(xii)\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image021.png\" \/><\/strong><\/p>\n<p>(xiii)\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image022.png\" \/><\/strong><\/p>\n<p>(xiv)\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image023.png\" \/><\/strong><\/p>\n<p>(xv)\u00a0\u00a0\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image024.png\" \/><\/strong><\/p>\n<p>(xvi)\u00a0\u00a0 <strong><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch07\/image025.png\" \/><\/strong><\/p>\n<ul>\n<li><strong>Integration by Partial Fraction<\/strong>: A rational fraction is the ratio of the form <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{{\\rm{P}}\\left( x \\right)} \\over {{\\rm{Q}}\\left( x \\right)}}\" \/><\/span><\/span>\u00a0where P(x) and Q(x) are polynomials in x and Q(x) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\ne\" \/><\/span><\/span>\u00a00. If the degree of the polynomial P(x) is greater than the degree of the polynomial Q(x), then we may divide P(x) by Q(x) so that <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{{\\rm{P}}\\left( x \\right)} \\over {{\\rm{Q}}\\left( x \\right)}} = {\\rm{T}}\\left( x \\right) + {{{{\\rm{P}}_1}\\left( x \\right)} \\over {{\\rm{Q}}\\left( x \\right)}}\" \/><\/span><\/span>\u00a0where T(x) is a polynomial in x and degree of P<sub>1<\/sub>(x) is less than Q(x). T(x) is a polynomial can be easily integrated.<\/li>\n<\/ul>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{{{\\rm{P}}_1}\\left( x \\right)} \\over {{\\rm{Q}}\\left( x \\right)}}\" \/><\/span><\/span>can be integrated by expressing <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{{{\\rm{P}}_1}\\left( x \\right)} \\over {{\\rm{Q}}\\left( x \\right)}}\" \/><\/span><\/span>\u00a0as the sum of partial fractions of the following types:<\/p>\n<p>(a) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{px + q} \\over {\\left( {x - a} \\right)\\left( {x - b} \\right)}} = {{\\rm{A}} \\over {x - a}} + {{\\rm{B}} \\over {x - b}}\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x \\ne a,x \\ne b,a \\ne b\" \/><\/span><\/span><\/p>\n<p>(b) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{px + q} \\over {{{\\left( {x - a} \\right)}^2}}} = {{\\rm{A}} \\over {x - a}} + {{\\rm{B}} \\over {{{\\left( {x - a} \\right)}^2}}}\" \/><\/span><\/span><\/p>\n<p>(c) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{p{x^2} + qx + r} \\over {\\left( {x - a} \\right)\\left( {x - b} \\right)\\left( {x - c} \\right)}} = {{\\rm{A}} \\over {x - a}} + {{\\rm{B}} \\over {x - b}} + {{\\rm{C}} \\over {x - c}}\" \/><\/span><\/span><\/p>\n<p>(d) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{p{x^2} + qx + r} \\over {{{\\left( {x - a} \\right)}^2}\\left( {x - b} \\right)}} = {{\\rm{A}} \\over {x - a}} + {{\\rm{B}} \\over {{{\\left( {x - a} \\right)}^2}}} + {{\\rm{C}} \\over {x - b}}\" \/><\/span><\/span><\/p>\n<p>(e) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{p{x^2} + qx + r} \\over {\\left( {x - a} \\right)\\left( {{x^2} + bx + c} \\right)}} = {{\\rm{A}} \\over {x - a}} + {{{\\rm{B}}x + {\\rm{C}}} \\over {{x^2} + bx + c}}\" \/><\/span><\/span>\u00a0where <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{x^2} + bx + c}\" \/><\/span><\/span>\u00a0cannot be factorize further into linear fraction.<\/p>\n<ul>\n<li><strong>Integration by Substitution<\/strong>: In this method we change the variable to some other variable. When the integrand involves some trigonometric functions, we shall be using some well-known identities to find the integrals. Using substitution technique, we obtain the following standard integrals:<\/li>\n<\/ul>\n<p>(a) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {\\tan x{\\rm{ }}dx} = \\log \\left| {\\sec x} \\right| + c\" \/><\/span><\/span><\/p>\n<p>(b) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {\\cot x{\\rm{ }}dx} = \\log \\left| {\\sin x} \\right| + c\" \/><\/span><\/span><\/p>\n<p>(c) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {\\sec x{\\rm{ }}dx} = \\log \\left| {\\sec x + \\tan x} \\right| + c\" \/><\/span><\/span><\/p>\n<p>(d) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {\\cos ec{\\rm{ }}x{\\rm{ }}dx} = \\log \\left| {\\cos ec{\\rm{ }}x - \\cot x} \\right| + c\" \/><\/span><\/span><\/p>\n<ul>\n<li><strong>Integrals of Some Special Functions<\/strong>:<\/li>\n<\/ul>\n<p>(a) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{dx} \\over {{x^2} - {a^2}}}} = {1 \\over {2a}}\\log \\left| {{{x - a} \\over {x + a}}} \\right| + c\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (b) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{dx} \\over {{a^2} - {x^2}}}} = {1 \\over {2a}}\\log \\left| {{{a + x} \\over {a - x}}} \\right| + c\" \/><\/span><\/span><\/p>\n<p>(c) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{dx} \\over {{x^2} + {a^2}}}} = {1 \\over a}{\\tan ^{ - 1}}{x \\over a} + c\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(d) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{dx} \\over {\\sqrt {{x^2} - {a^2}} }}} = \\log \\left| {x + \\sqrt {{x^2} - {a^2}} } \\right| + c\" \/><\/span><\/span><\/p>\n<p>(e) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{dx} \\over {\\sqrt {{a^2} + {x^2}} }}} = \\log \\left| {x + \\sqrt {{a^2} + {x^2}} } \\right| + c\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(f)\u00a0 <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{dx} \\over {\\sqrt {{a^2} - {x^2}} }}} = {\\sin ^{ - 1}}{x \\over a} + c\" \/><\/span><\/span><\/p>\n<ul>\n<li><strong>Integration by Parts<\/strong>: For the given function f(x) and g(x),<\/li>\n<\/ul>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {f\\left( x \\right).g\\left( x \\right){\\rm{ }}dx} = f\\left( x \\right)\\int {g\\left( x \\right){\\rm{ }}dx} - \\int {\\left\\{ {f'\\left( x \\right)\\int {g\\left( x \\right){\\rm{ }}dx} } \\right\\}{\\rm{ }}dx}\" \/><\/span><\/span><\/p>\n<p>We must take proper care to choose the first function and second function clearly. We must choose that function as the second function whose integral is well-known to us.<\/p>\n<ul>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{e^x}\\left[ {f\\left( x \\right) + f'\\left( x \\right)} \\right]{\\rm{ }}dx} = {e^x}f\\left( x \\right) + c\" \/><\/span><\/span><\/li>\n<li><strong>Some Special Type of Integrals<\/strong>:<\/li>\n<\/ul>\n<p>(a) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {\\sqrt {{a^2} - {x^2}} {\\rm{ }}dx} = {1 \\over 2}\\left[ {x\\sqrt {{a^2} - {x^2}} + {a^2}\\sin {x \\over a}} \\right]\" \/><\/span><\/span><\/p>\n<p>(b) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {\\sqrt {{x^2} + {a^2}} {\\rm{ }}dx} = {1 \\over 2}\\left[ {x\\sqrt {{x^2} + {a^2}} + {a^2}\\log \\left| {x + \\sqrt {{x^2} + {a^2}} } \\right|} \\right] + c\" \/><\/span><\/span><\/p>\n<p>(c) <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {\\sqrt {{x^2} - {a^2}} {\\rm{ }}dx} = {1 \\over 2}\\left[ {x\\sqrt {{x^2} - {a^2}} + {a^2}\\log \\left| {x + \\sqrt {{x^2} - {a^2}} } \\right|} \\right] + c\" \/><\/span><\/span><\/p>\n<ul>\n<li><strong>Integrals of the types <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{dx} \\over {a{x^2} + bx + c}}}\" \/><\/span><\/span><\/strong>\u00a0or <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{dx} \\over {\\sqrt {a{x^2} + bx + c} }}}\" \/><\/span><\/span><\/li>\n<\/ul>\n<p>(a) These type of integrals are transformed into standard form by expressing<\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a{x^2} + bx + c = a\\left( {{x^2} + {{bx} \\over a} + {c \\over a}} \\right) = a\\left[ {{{\\left( {x + {b \\over {2a}}} \\right)}^2} + \\left( {{c \\over a} - {{{b^2}} \\over {4{a^2}}}} \\right)} \\right]\" \/><\/span><\/span><\/p>\n<p>(b) Integrals of the types <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{px + q} \\over {a{x^2} + bx + c}}{\\rm{ }}dx}\" \/><\/span><\/span>\u00a0or <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int {{{px + q} \\over {\\sqrt {a{x^2} + bx + c} }}{\\rm{ }}dx}\" \/><\/span><\/span>\u00a0are transformed into standard form by expressing <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?px + d = {\\rm{A}}{d \\over {dx}}\\left( {a{x^2} + bx + c} \\right) + {\\rm{B}}\" \/><\/span><\/span><\/p>\n<p><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\Rightarrow\" \/><\/span><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?px + d = {\\rm{A}}\\left( {2ax + b} \\right) + {\\rm{B}}\" \/><\/span><\/span><\/p>\n<p>where A abd B are determined by comparing coefficients on both sides.<\/p>\n<ul>\n<li>We have already defined <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int\\limits_a^b {f\\left( x \\right){\\rm{ }}dx}\" \/><\/span><\/span>\u00a0as the area of the region bounded by the curve <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y = f\\left( x \\right)\" \/><\/span><\/span>, <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a \\le x \\le b,\" \/><\/span><\/span>\u00a0the x-axis and the ordinates x = a and x = b. Let x be a given point in [a, b], then <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int\\limits_a^b {f\\left( x \\right){\\rm{ }}dx}\" \/><\/span><\/span>\u00a0represents the area function A(x).<\/li>\n<li>First Fundamental Theorem of Integral Calculus: Let the area function be defined by <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{A}}\\left( x \\right) = \\int\\limits_a^b {f\\left( x \\right){\\rm{ }}dx}\" \/><\/span><\/span>\u00a0for all <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x \\ge a\" \/><\/span><\/span>\u00a0where the function f is assumed to be continuous on [a, b], then <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{A'}}\\left( x \\right) = f\\left( x \\right)\" \/><\/span><\/span>\u00a0for all <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x \\in \\left[ {a,b} \\right]\" \/><\/span><\/span>.<\/li>\n<li>Secind Fundamental Theorem of Integral Calculus: Let f be a continuous function of x defined on the closed interval ]a, b] and let F be another function such that <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}{\\rm{F}}\\left( x \\right) = f\\left( x \\right)\" \/><\/span><\/span>\u00a0for all x in the domain of f, then <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\int\\limits_a^b {f\\left( x \\right){\\rm{ }}dx} = {\\rm{F}}\\left( x \\right) + \\left. {\\rm{C}} \\right]_a^b = {\\rm{F}}\\left( b \\right) - {\\rm{F}}\\left( a \\right)\" \/><\/span><\/span><\/li>\n<\/ul>\n<p>This is called the definite integral of f over the range [a, b] were a and b are called the limits of integration,\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>\u00a0being the lower limit and\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?b\" \/><\/span><\/span>\u00a0the upper limit.<strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><\/p>\n<h2><strong>CBSE Class CBSE Revision notes (PDF Download) Free<br \/>\nCBSE Revision notes for Class 12 Mathematics PDF<br \/>\nCBSE Revision notes Class 12 Mathematics \u2013 CBSE<br \/>\nCBSE Revisions notes and Key Points Class 12 Mathematics<br \/>\nSummary of the NCERT books all chapters in Mathematics class 12<br \/>\nShort notes for CBSE class 12th Mathematics<br \/>\nKey notes and chapter summary of Mathematics class 12<br \/>\nQuick revision notes for CBSE board exams<br \/>\n12 Revision Notes and Key Points<\/strong><\/h2>\n<p>Integrals class 12 Notes Mathematics. CBSE quick revision note for class-12 Chemistry Physics Math\u2019s, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.<\/p>\n<ul>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-physics\/1251\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Physics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-chemistry\/1267\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Chemistry<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics\/1284\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Mathematics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-biology\/1298\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Biology<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-accountancy\/1315\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Accountancy<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-economics\/1327\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Economics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-business-studies\/1727\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Business Studies<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-computer-science\/1851\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Computer Science<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-informatics-practices\/1873\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Informatics Practices<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-english-core\/1855\/cbse-revision-notes\/7\/\">Revision Notes for class-12 English Core<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-history\/1869\/cbse-revision-notes\/7\/\">Revision Notes for class-12 History<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-physical-education\/1877\/cbse-revision-notes\/7\/\">Revision Notes for class-12 Physical Education<\/a><\/li>\n<\/ul>\n<p>To download Integrals class 12 Notes Mathematics, sample paper for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT Integrals, NCERT Exemplar Integrals, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n<ul>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/\">Relations and Functions class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/inverse-trigonometric-functions-class-12-notes-mathematics\/\">Inverse Trigonometric Functions class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/matrices-class-12-notes-mathematics\/\">Matrices class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/determinants-class-12-notes-mathematics\/\">Determinants class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/continuity-differentiability-class-12-notes-mathematics\/\">Continuity and Differentiability class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/application-derivatives-class-12-notes-mathematics\/\">Application of Derivatives class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/application-integrals-class-12-notes-mathematics\/\">Application of Integrals class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/differential-equations-class-12-notes-mathematics\/\">Differential Equations class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/vector-algebra-class-12-notes-mathematics\/\">Vector Algebra class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/three-dimensional-geometry-class-12-notes-mathematics\/\">Three Dimensional Geometry class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/linear-programming-class-12-notes-mathematics\/\">Linear Programming class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/\">Probability class 12 Notes Mathematics<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Integrals Class 12 Notes Mathematics in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Integrals class 12 Notes latest chapter wise notes for quick preparation of CBSE board exams and school-based annual examinations. Class 12 Mathematics notes on chapter 7 Integrals are also available for &#8230; <a title=\"Integrals Class 12 Notes Mathematics\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/integrals-class-12-notes-mathematics\/\" aria-label=\"More on Integrals Class 12 Notes Mathematics\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":9266,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[48,456],"tags":[457,150,565,461,426,240],"class_list":["post-9324","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cbse-class-12","category-revision-notes","tag-cbse-notes","tag-cbse-notes-and-key-points","tag-integrals","tag-mathematics-notes","tag-quick-revision","tag-quick-revision-notes"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Integrals Class 12 Notes Mathematics | myCBSEguide<\/title>\n<meta name=\"description\" content=\"Integrals class 12 Notes Mathematics chapter 7 in PDF format for free download. 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