{"id":71655,"date":"2022-02-07T17:00:49","date_gmt":"2022-02-07T11:30:49","guid":{"rendered":"https:\/\/www.cbselabs.com\/?p=71655"},"modified":"2022-02-07T17:11:10","modified_gmt":"2022-02-07T11:41:10","slug":"integrals-class-12-notes","status":"publish","type":"post","link":"https:\/\/www.cbselabs.com\/integrals-class-12-notes\/","title":{"rendered":"Integrals Class 12 Notes Maths Chapter 7"},"content":{"rendered":"<p><a href=\"https:\/\/www.cbselabs.com\/cbse-notes-class-12-maths\/\">Class 12 Maths Notes<\/a> students can refer to the Integrals Class 12 Notes Maths Chapter 7 https:\/\/www.cbselabs.com\/integrals-class-12-notes\/ Pdf here. They can also access the CBSE Class 12 Integrals Chapter 7 Notes while gearing up for their Board exams.<\/p>\n<h2>CBSE Class 12 Maths Notes Chapter 7 Integrals<\/h2>\n<p><strong>Integrals Class 12 Notes Chapter 7<\/strong><\/p>\n<p>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.<br \/>\nThen, \u222bf(x) dx = F(x) + C, these integrals are called indefinite integrals or general integrals. C is an arbitrary constant by varying which one gets different anti-derivatives of the given function.<br \/>\nNote: Derivative of a function is unique but a function can have infinite anti-derivatives or integrals.<\/p>\n<p><strong>Integration Class 12 Notes Chapter 7<\/strong><\/p>\n<p><strong>Properties of Indefinite Integral<\/strong><br \/>\n(i) \u222b[f(x) + g(x)] dx = \u222bf(x) dx + \u222bg(x) dx<br \/>\n(ii) For any real number k, \u222bk f(x) dx = k\u222bf(x)dx.<br \/>\n(iii) In general, if f<sub>1<\/sub>, f<sub>2<\/sub>,&#8230;&#8230;&#8230;, f<sub>n<\/sub> are functions and k<sub>1<\/sub>, k<sub>2<\/sub>,&#8230;, k<sub>n<\/sub> are real numbers, then<br \/>\n\u222b[k<sub>1<\/sub>f<sub>1<\/sub>(x) + k<sub>2<\/sub> f<sub>2<\/sub>(x)+&#8230;+ k<sub>n<\/sub>f<sub>n<\/sub>(x)] dx = k<sub>1<\/sub> \u222bf<sub>1<\/sub>(x) dx + k<sub>2<\/sub> \u222b f<sub>2<\/sub>(x) dx+&#8230;+ k<sub>n<\/sub> \u222bf<sub>n<\/sub>(x) dx<\/p>\n<p><strong>Basic Formulae<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone\" src=\"https:\/\/i1.wp.com\/live.staticflickr.com\/65535\/46990633065_67315efd43_o.png?resize=678%2C291&#038;ssl=1\" alt=\"Integrals Class 12 Notes Chapter 7\" width=\"678\" height=\"291\" data-recalc-dims=\"1\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone\" src=\"https:\/\/i0.wp.com\/live.staticflickr.com\/65535\/40940177933_889abec611_o.png?resize=689%2C604&#038;ssl=1\" alt=\"Integration Class 12 Notes Chapter 7\" width=\"689\" height=\"604\" data-recalc-dims=\"1\" \/><\/p>\n<p><strong>Integration Notes Chapter 7 Class 12<\/strong><\/p>\n<p><strong>Integration using Trigonometric Identities<\/strong><br \/>\nWhen the integrand involves some trigonometric functions, we use the following identities to find the integral:<\/p>\n<ul>\n<li>2 sin A . cos B = sin( A + B) + sin( A &#8211; B)<\/li>\n<li>2 cos A . sin B = sin( A + B) &#8211; sin( A &#8211; B)<\/li>\n<li>2 cos A . cos B = cos (A + B) + cos(A &#8211; B)<\/li>\n<li>2 sin A . sin B = cos(A &#8211; B) &#8211; cos (A + B)<\/li>\n<li>2 sin A cos A = sin 2A<\/li>\n<li>cos<sup>2<\/sup> A &#8211; sin<sup>2<\/sup> A = cos 2A<\/li>\n<li>sin<sup>2<\/sup> A = (\\(\\frac { 1-cos2A }{ 2 }\\))<\/li>\n<li>sin<sup>2<\/sup> A + cos<sup>2<\/sup> A = 1<\/li>\n<li>\\({ sin }^{ 3 }A=\\frac { 3sinA-sin3A }{ 4 }\\)<\/li>\n<li>\\({ cos }^{ 3 }A=\\frac { 3cosA+cos3A }{ 4 }\\)<\/li>\n<\/ul>\n<p><strong>Integration Notes Class 12 Chapter 7<\/strong><\/p>\n<p><strong>Integration by Substitutions<\/strong><br \/>\nSubstitution method is used, when a suitable substitution of variable leads to simplification of integral.<br \/>\nIf I = \u222bf(x)dx, then by putting x = g(z), we get<br \/>\nI = \u222b f[g(z)] g'(z) dz<br \/>\nNote: Try to substitute the variable whose derivative is present in the original integral and final integral must be written in terms of the original variable of integration.<\/p>\n<p><strong>Integration by Parts<\/strong><br \/>\nFor a given functions f(x) and q(x), we have<br \/>\n\u222b[f(x) q(x)] dx = f(x)\u222bg(x)dx &#8211; \u222b{f'(x) \u222bg(x)dx} dx<br \/>\nHere, we can choose the first function according to its position in ILATE, where<br \/>\nI = Inverse trigonometric function<br \/>\nL = Logarithmic function<br \/>\nA = Algebraic function<br \/>\nT = Trigonometric function<br \/>\nE = Exponential function<br \/>\n[the function which comes first in ILATE should taken as first junction and other as second function]<\/p>\n<p>Note<br \/>\n(i) Keep in mind, ILATE is not a rule as all questions of integration by parts cannot be done by above method.<br \/>\n(ii) It is worth mentioning that integration by parts is not applicable to product of functions in all cases. For instance, the method does not work for \u222b\u221ax sinx dx. The reason is that there does not exist any function whose derivative is \u221ax sinx.<br \/>\n(iii) Observe that while finding the integral of the second function, we did not add any constant of integration.<\/p>\n<p><strong>Class 12 Maths Chapter 7 Notes<\/strong><\/p>\n<p><strong>Integration by Partial Fractions<\/strong><br \/>\nA rational function is ratio of two polynomials of the form \\(\\frac { p(x) }{ q(x) }\\), where p(x) and q(x) are polynomials in x and q(x) \u2260 0. If degree of p(x) &gt; degree of q(x), then we may divide p(x) by q(x) so that \\(\\frac { p(x) }{ q(x) } =t(x)+\\frac { { p }_{ 1 }(x) }{ q(x) }\\), where t(x) is a polynomial in x which can be integrated easily and degree of p1(x) is less than the degree of q(x) . \\(\\frac { { p }_{ 1 }(x) }{ q(x) }\\) can be integrated by expressing \\(\\frac { { p }_{ 1 }(x) }{ q(x) }\\) as the sum of partial fractions of the following type:<br \/>\n<img loading=\"lazy\" class=\"alignnone\" src=\"https:\/\/i2.wp.com\/live.staticflickr.com\/65535\/40940178013_498a9e4ec5_o.png?resize=277%2C105&#038;ssl=1\" alt=\"Integration Notes Chapter 7 Class 12\" width=\"277\" height=\"105\" data-recalc-dims=\"1\" \/><br \/>\n<img loading=\"lazy\" class=\"alignnone\" src=\"https:\/\/i1.wp.com\/live.staticflickr.com\/65535\/47854616962_fdb64fd910_o.png?resize=338%2C164&#038;ssl=1\" alt=\"Integration Notes Class 12 Chapter 7\" width=\"338\" height=\"164\" data-recalc-dims=\"1\" \/><br \/>\nwhere x<sup>2<\/sup> + bx + c cannot be factorised further.<\/p>\n<p>Integrals of the types <img loading=\"lazy\" class=\"alignnone\" src=\"https:\/\/i0.wp.com\/live.staticflickr.com\/65535\/40940177843_1eba9ec178_o.png?resize=240%2C55&#038;ssl=1\" alt=\"Class 12 Maths Chapter 7 Notes\" width=\"240\" height=\"55\" data-recalc-dims=\"1\" \/> can be transformed into standard form by expressing <img loading=\"lazy\" class=\"alignnone\" src=\"https:\/\/i0.wp.com\/live.staticflickr.com\/65535\/46990632935_bd5d9d1c0c_o.png?resize=411%2C68&#038;ssl=1\" alt=\"Class 12 Integration Notes Chapter 7\" width=\"411\" height=\"68\" data-recalc-dims=\"1\" \/><\/p>\n<p>Integrals of the types <img loading=\"lazy\" class=\"alignnone\" src=\"https:\/\/i2.wp.com\/live.staticflickr.com\/65535\/40940177733_b8879dcbd5_o.png?resize=294%2C56&#038;ssl=1\" alt=\"Integrals Notes Class 12 Chapter 7\" width=\"294\" height=\"56\" data-recalc-dims=\"1\" \/> can be transformed into standard form by expressing px + q = A \\(\\frac { d }{ dx }\\) (ax<sup>2<\/sup> + bx + c) + B = A(2ax + b) + B, where A and B are determined by comparing coefficients on both sides.<\/p>\n<h5><a href=\"https:\/\/www.cbselabs.com\/ncert-solutions-2\/\">NCERT Solutions<\/a><\/h5>\n","protected":false},"excerpt":{"rendered":"<p>Class 12 Maths Notes students can refer to the Integrals Class 12 Notes Maths Chapter 7 https:\/\/www.cbselabs.com\/integrals-class-12-notes\/ Pdf here. They can also access the CBSE Class 12 Integrals Chapter 7 Notes while gearing up for their Board exams. CBSE Class 12 Maths Notes Chapter 7 Integrals Integrals Class 12 Notes Chapter 7 Integration is the &hellip;<\/p>\n","protected":false},"author":27,"featured_media":164611,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[2],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.2 (Yoast SEO v17.2) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Integrals Class 12 Notes Maths Chapter 7 - CBSE Labs<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.cbselabs.com\/integrals-class-12-notes\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integrals Class 12 Notes Maths Chapter 7\" \/>\n<meta property=\"og:description\" content=\"Class 12 Maths Notes students can refer to the Integrals Class 12 Notes Maths Chapter 7 https:\/\/www.cbselabs.com\/integrals-class-12-notes\/ Pdf here. 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They can also access the CBSE Class 12 Integrals Chapter 7 Notes while gearing up for their Board exams. 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