{"id":69141,"date":"2021-08-26T19:31:00","date_gmt":"2021-08-26T14:01:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=69141"},"modified":"2021-09-02T18:10:10","modified_gmt":"2021-09-02T12:40:10","slug":"rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/","title":{"rendered":"RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31 (2021)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-120091\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Solutions-Class-12-Chapter-19-Exercise-19.31.png\" alt=\"RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31\" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Solutions-Class-12-Chapter-19-Exercise-19.31.png 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Solutions-Class-12-Chapter-19-Exercise-19.31-768x432.png 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><strong>RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31<\/strong>: Solutions PDF is an important reference guide to help students score well in the Class 12 exam. By solving exercise problems on a daily basis, students improve their problem solving and logical thinking skills. The main goal is to help students self-analyze the areas that require the most practice from the exam point of view. With the help of RD Sharma solutions, students can solve exercise problems in a short duration clearly.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-12-solutions\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions For Class 12th\u00a0Maths All Chapters<\/a><\/li>\n<\/ul>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_47_1 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"ez-toc-toggle-icon-1\"><label for=\"item-69ea6a0121d6c\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input  type=\"checkbox\" id=\"item-69ea6a0121d6c\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-visibility-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#download-rd-sharma-solutions-class-12-maths-chapter-19-exercise-1931-pdf\" title=\"Download RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31 PDF:\">Download RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31 PDF:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#access-rd-sharma-solutions-class-12-maths-chapter-19-exercise-1931\" title=\"Access RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31\">Access RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#rd-sharma-solutions-class-12-maths-chapter-19-exercise-1931-important-topics\" title=\"RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31: Important Topics\">RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31: Important Topics<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#faq-rd-sharma-solutions-class-12-maths-chapter-19-exercise-1931\" title=\"FAQ: RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31\">FAQ: RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#what-are-indefinite-integrals\" title=\"What are indefinite integrals? \">What are indefinite integrals? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#how-indefinite-integrals-are-represented\" title=\"How indefinite integrals are represented? \">How indefinite integrals are represented? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#what-is-the-indefinite-integral-of-0\" title=\"What is the indefinite integral of 0?\">What is the indefinite integral of 0?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#are-integrals-tough\" title=\"Are integrals tough? \">Are integrals tough? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/#what-is-the-relation-between-indefinite-integral-and-antiderivative\" title=\"What is the relation between indefinite integral and antiderivative? \">What is the relation between indefinite integral and antiderivative? <\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-solutions-class-12-maths-chapter-19-exercise-1931-pdf\"><\/span>Download RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31 PDF:<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div id=\"example1\" style=\"text-align: justify;\">\u00a0<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 500px;}<br \/>\n.pdfobject { border: 1px solid #666; }<br \/>\n<\/style><\/p>\n<p style=\"text-align: justify;\"><script src=\"https:\/\/www.kopykitab.com\/_utility\/js\/pdfobject.min.js\"><\/script><br \/><script>PDFObject.embed(\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-12-Maths-Chapter-19-Ex-19.31.pdf\", \"#example1\");<\/script><\/p>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-12-Maths-Chapter-19-Ex-19.31.pdf\">RD-Sharma-Solutions-Class-12-Maths-Chapter-19-Ex-19.31<\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"access-rd-sharma-solutions-class-12-maths-chapter-19-exercise-1931\"><\/span>Access RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Indefinite Integrals Ex 19.31 Q1<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/1\/634\/32099900143_67e02cb5f5_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q1\" width=\"409\" height=\"423\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q2<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/1\/399\/32874243606_8381d7f199_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q2\" width=\"501\" height=\"739\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q3<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/3\/2343\/32099899833_6f8d030e8d_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q3\" width=\"402\" height=\"408\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q4<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/1\/380\/32874243456_3523777c4b_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q4\" width=\"422\" height=\"608\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q5<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/3\/2933\/32874243426_a1f9766fa4_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q5\" width=\"410\" height=\"655\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q6<\/strong><br \/><img src=\"https:\/\/c2.staticflickr.com\/4\/3724\/32874243286_f6b4e0812a_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q6\" width=\"403\" height=\"415\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q7<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/3\/2912\/32874243216_3a73856bd9_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q7\" width=\"358\" height=\"472\" \/><\/p>\n<p><strong>Indefinite Integrals Ex 19.31 Q8<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/3\/2117\/32874243176_d710c07700_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q8\" width=\"403\" height=\"469\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q9<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/3\/2170\/32099899483_3f7532c010_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q9\" width=\"404\" height=\"698\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q10<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/1\/730\/32874243066_c654fa07f1_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q10\" width=\"407\" height=\"608\" \/><br \/><strong>Indefinite Integrals Ex 19.31 Q11<\/strong><br \/><img src=\"https:\/\/c1.staticflickr.com\/1\/632\/32874242996_14efe1c085_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q11\" width=\"524\" height=\"776\" \/><br \/><img src=\"https:\/\/c1.staticflickr.com\/3\/2081\/32874243686_b345ee8b0d_o.png\" alt=\"RD-Sharma-Class-12-Solutions-Chapter-19-indefinite-integrals-Ex-19.31-Q11-1\" width=\"400\" height=\"360\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"rd-sharma-solutions-class-12-maths-chapter-19-exercise-1931-important-topics\"><\/span>RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31: Important Topics<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Some of the essential topics covered in\u00a0<a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-indefinite-integrals\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 12 Maths Chapter 19<\/a> Exercise 19.31 chapter are listed below.<\/p>\n<ul>\n<li>Definition of primitive or antiderivative<\/li>\n<li>Definition and meaning of indefinite integral<\/li>\n<li>Fundamental integration formulae<\/li>\n<li>Some standard results on integration along with the corollary<\/li>\n<li>Integration of trigonometric functions<\/li>\n<li>Integration of exponential functions<\/li>\n<li>Miscellaneous problems<\/li>\n<li>Geometrical interpretation of indefinite integral<\/li>\n<li>Comparison between differentiation and integration<\/li>\n<li>Methods of integration<\/li>\n<li>Integration by substitution<\/li>\n<li>Some standard results<\/li>\n<li>Evaluation of integrals by using trigonometric substitutions<\/li>\n<li>Some special integrals<\/li>\n<li>Integration by parts<\/li>\n<li>Some important integrals along with theorems<\/li>\n<li>Integration of rational algebraic functions by using partial fractions\n<ul>\n<li>When the denominator is expressible as a product of distinct linear factors<\/li>\n<li>When the denominator contains some repeating linear factors<\/li>\n<li>The denominator contains irreducible quadratic factors<\/li>\n<\/ul>\n<\/li>\n<li>Integration of some special irrational algebraic functions<\/li>\n<\/ul>\n<p>We have included all the information regarding\u00a0<a href=\"https:\/\/www.cbse.gov.in\/\" target=\"_blank\" rel=\"noopener\">CBSE<\/a> RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31. If you ahve any query feel free to ask in the comment section.\u00a0<\/p>\n<h3><span class=\"ez-toc-section\" id=\"faq-rd-sharma-solutions-class-12-maths-chapter-19-exercise-1931\"><\/span>FAQ: RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1629985116382\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"what-are-indefinite-integrals\"><\/span>What are indefinite integrals? <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>An indefinite integral is\u00a0a function that takes the antiderivative of another function.\u00a0<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629985152153\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"how-indefinite-integrals-are-represented\"><\/span>How indefinite integrals are represented? <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>\u00a0It is visually represented as an integral symbol, a function, and then a dx at the end.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629985155541\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"what-is-the-indefinite-integral-of-0\"><\/span>What is the indefinite integral of 0?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>The integral of 0 is\u00a0C because the derivative of C is zero.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629985156564\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"are-integrals-tough\"><\/span>Are integrals tough? <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Yes, students find integrals tough than derivatives. <\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629985157776\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"what-is-the-relation-between-indefinite-integral-and-antiderivative\"><\/span>What is the relation between indefinite integral and antiderivative? <span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>The indefinite integral is an easier way to symbolize taking the antiderivative.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31: Solutions PDF is an important reference guide to help students score well in the Class 12 exam. By solving exercise problems on a daily basis, students improve their problem solving and logical thinking skills. The main goal is to help students self-analyze the areas that &#8230; <a title=\"RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31 (2021)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-19-exercise-19-31\/\" aria-label=\"More on RD Sharma Solutions Class 12 Maths Chapter 19 Exercise 19.31 (2021)\">Read more<\/a><\/p>\n","protected":false},"author":244,"featured_media":120091,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[2917,73716,73411,2985,73410],"tags":[3429,73223,73664,76761],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/69141"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/244"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=69141"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/69141\/revisions"}],"predecessor-version":[{"id":123925,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/69141\/revisions\/123925"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/120091"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=69141"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=69141"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=69141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}