{"id":31744,"date":"2024-02-08T17:41:44","date_gmt":"2024-02-08T12:11:44","guid":{"rendered":"https:\/\/ncertsolutions.guru\/?p=31744"},"modified":"2024-02-09T17:41:09","modified_gmt":"2024-02-09T12:11:09","slug":"ncert-solutions-for-class-12-maths-chapter-7-ex-7-2","status":"publish","type":"post","link":"https:\/\/ncertsolutions.guru\/ncert-solutions-for-class-12-maths-chapter-7-ex-7-2\/","title":{"rendered":"NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2"},"content":{"rendered":"

These NCERT Solutions for Class 12 Maths<\/a> Chapter 7 Integrals Ex 7.2 Questions and Answers are prepared by our highly skilled subject experts.<\/p>\n

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.2<\/h2>\n

\"NCERT<\/p>\n

Question 1.
\n\\(\\frac { 2x }{ 1+{ x }^{ 2 } }\\)
\nSolution:
\nLet 1 + x\u00b2 = t
\n\u21d2 2xdx = dt
\n\u2234 \u222b\\(\\frac { 2x }{ 1+{ x }^{ 2 } }\\) dx
\n= \u222b\\(\\frac { dt }{ t }\\)
\n= log t + C
\n= log(1 + x\u00b2) + C<\/p>\n

Question 2.
\n\\(\\frac { { \\left( logx \\right) }^{ 2 } }{ x } \\)
\nSolution:
\nLet logx = t
\n\u21d2 \\(\\frac { 1 }{ x }dx=dt\\)
\n\\(\\int \\frac{(\\log x)^{2}}{x} d x=\\int t^{2} d t=\\frac{t^{3}}{3}+\\mathrm{C}\\)
\n= \\(\\frac{1}{3}(\\log x)^{3}+\\mathrm{C}\\)<\/p>\n

Question 3.
\n\\(\\frac { 1 }{ x+xlogx }\\)
\nSolution:
\nPut 1 + logx = t
\n\u2234 \\(\\frac { 1 }{ x }\\)dx = dt
\n\\(\\int { \\frac { 1 }{ x(1+logx) } dx } =\\int { \\frac { 1 }{ t } dt } =log|t| + C \\)
\n= log|1+logx| + C<\/p>\n

\"NCERT<\/p>\n

Question 4.
\nsinx sin(cosx)
\nSolution:
\n\u222bsinx sin(cosx)dx = \u222bsin t dt
\n= – (-cos t) + C
\n= – cost + C
\n= cos(cos x) + C<\/p>\n

Question 5.
\nsin(ax+b) cos(ax+b)
\nSolution:
\n\"NCERT
\nAnother method:
\n\"NCERT<\/p>\n

Question 6.
\n\\(\\sqrt { ax+b }\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 7.
\nx\\(\\sqrt { x+2 }\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 8.
\nx\\(\\sqrt { 1+{ 2x }^{ 2 } }\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 9.
\n(4x+2)\\(\\sqrt { { x }^{ 2 }+ x + 1 } \\)
\nSolution:
\n\"NCERT<\/p>\n

\"NCERT<\/p>\n

Question 10.
\n\\(\\frac { 1 }{ x-\\sqrt { x } } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 11.
\n\\(\\frac { x }{ \\sqrt { x+4 } }\\) , x > 0
\nSolution:
\n\"NCERT<\/p>\n

Question 12.
\n\\({ { (x }^{ 3 }-1) }^{ \\frac { 1 }{ 3 } }.{ x }^{ 5 }\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 13.
\n\\(\\frac { { x }^{ 2 } }{ { { (2+3x }^{ 3 }) }^{ 3 } } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 14.
\n\\(\\frac { 1 }{ x(logx)^{ m } }\\), x > 0
\nSolution:
\nPut log x = t, \\(\\frac { dt }{ dx }\\) = \\(\\frac { 1 }{ x }\\) \u21d2 dt = \\(\\frac { 1 }{ x }\\)dx
\n\\(\\int \\frac{1}{x(\\log x)^{m}} d x=\\int \\frac{d t}{t^{m}}=\\int t^{-m} d t=\\frac{t^{-m+1}}{-m+1}+\\mathrm{C}\\)
\n\\(=\\frac { { (logx) }^{ 1-m } }{ 1-m }\\) + C<\/p>\n

\"NCERT<\/p>\n

Question 15.
\n\\(\\frac { x }{ 9-4{ x }^{ 2 } } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 16.
\n\\({ e }^{ 2x+3 }\\)
\nSolution:
\nput 2x + 3 = t
\nso that 2dx = dt
\n\\(\\int { { e }^{ 2x+3 } } dx\\quad =\\frac { 1 }{ 2 } \\int { { e }^{ t }dt } \\quad =\\frac { 1 }{ 2 } { e }^{ t }+c\\quad =\\frac { 1 }{ 2 } { e }^{ 2x+3 }\\) + C<\/p>\n

Question 17.
\n\\(\\frac { x }{ { e }^{ { x }^{ 2 } } } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 18.
\n\\(\\frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } } \\)
\nSolution:
\nlet \\(\\quad { tan }^{ -1 }x=t\\Rightarrow \\frac { 1 }{ 1+{ x }^{ 2 } } dx=dt\\)
\n\u2234 \\(\\frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } } \\)dx
\n= \u222bet<\/sup> dt
\n= et<\/sup>+C
\n= etan-1 <\/sup>+ C<\/p>\n

Question 19.
\n\\(\\frac { { e }^{ 2x }-1 }{ { e }^{ 2x }+1 } \\)
\nSolution:
\n\"NCERT
\nAnother method
\n\"NCERT<\/p>\n

Question 20.
\n\\(\\frac { { e }^{ 2x }-{ e }^{ 2x } }{ { e }^{ 2x }+{ e }^{ -2x } } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 21.
\ntan\u00b2(2x – 3)
\nSolution:
\n\u222btan\u00b2(2x – 3)dx = \u222b[sec\u00b2(2x – 3) – 1]dx = I
\nput 2x – 3 = t
\nso that 2dx = dt
\nI = \\(\\frac { 1 }{ 2 }\\) \u222bsec\u00b2t dt – x + C
\n= \\(\\frac { 1 }{ 2 }\\)t – x + C
\n= \\(\\frac { 1 }{ 2 }\\)tan(2x -3 ) – x + C<\/p>\n

\"NCERT<\/p>\n

Question 22.
\nsec\u00b2(7 – 4x)
\nSolution:
\n\u222bsec\u00b2(7 – 4x)dx
\n= \\(\\frac { tan(7-4x) }{ -4 }\\) + C<\/p>\n

Question 23.
\n\\(\\frac { { sin }^{ -1 }x }{ \\sqrt { 1-{ x }^{ 2 } } } \\)
\nSolution:
\nLet \\(\\quad { sin }^{ -1 }x=t\\quad\\Rightarrow\\frac { 1dx }{ \\sqrt { 1-{ x }^{ 2 } } }\\) = dt
\n\\(\\int { \\frac { { sin }^{ -1 }x }{ \\sqrt { 1-{ x }^{ 2 } } } dx } =\\int { t\\quad dt } =\\frac { 1 }{ 2 } { t }^{ 2 }+c=\\frac { 1 }{ 2 } { { (sin }^{ -1 }x) }^{ 2 }+c\\)<\/p>\n

Question 24.
\n\\(\\frac { 2cosx-3sinx }{ 6cosx+4sinx }\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 25.
\n\\(\\frac { 1 }{ { cos }^{ 2 }x{ (1-tanx) }^{ 2 } } \\)
\nSolution:
\n\"NCERT<\/p>\n

\"NCERT<\/p>\n

Question 26.
\n\\(\\frac { cos\\sqrt { x } }{ \\sqrt { x } } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 27.
\n\\(\\sqrt { sin2x } cos2x\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 28.
\n\\(\\frac { cosx }{ \\sqrt { 1+sinx } } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 29.
\ncotx log sinx
\nSolution:
\n\"NCERT<\/p>\n

Question 30.
\n\\(\\frac { sinx }{ 1+cosx }\\)
\nSolution:
\nput 1 + cosx = t
\n\u21d2 – sinx dx = dt
\n\u222b\\(\\frac { sinx }{ 1+cosx }\\)dx
\n= – \\(\\frac { dt }{ t }\\)
\n= – log|t| + c
\n= – log|1 + cosx| + C
\n= \\(\\log \\left|\\frac{1}{1+\\cos x}\\right|+\\mathrm{C}\\)<\/p>\n

\"NCERT<\/p>\n

Question 31.
\n\\(\\frac { sinx }{ { (1+cosx) }^{ 2 } } \\)
\nSolution:
\nput 1 + cosx = t
\nso that – sinx dx = dt
\n\u222b\\(\\frac { sinx }{ { (1+cosx) }^{ 2 } } \\) = – \\(\\int \\frac{d t}{t^{2}}=\\frac{1}{t}+C\\)
\n= \\(\\frac { 1 }{ t }\\) + C
\n= \\(\\frac { 1 }{ 1 + cosx }\\) + C<\/p>\n

Question 32.
\n\\(\\frac { 1 }{ 1 + cotx }\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 33.
\n\\(\\frac { 1 }{ 1-tanx }\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 34.
\n\\(\\frac { \\sqrt { tanx } }{ sinxcosx } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 35.
\n\\(\\frac { { (1+logx) }^{ 2 } }{ x } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 36.
\n\\(\\frac { (x+1){ (x+logx) }^{ 2 } }{ x } \\)
\nSolution:
\n\"NCERT<\/p>\n

\"NCERT<\/p>\n

Question 37.
\n\\(\\frac { { x }^{ 3 }sin({ tan }^{ -1 }{ x }^{ 4 }) }{ 1+{ x }^{ 8 } } dx \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 38.
\n\\(\\int { \\frac { { 10x }^{ 9 }+{ 10 }^{ x }log{ e }^{ 10 } }{ { x }^{ 10 }+{ 10 }^{ x } } dx } \\) equals
\n(a) 10x<\/sup> – x10<\/sup> + C
\n(b) 10x<\/sup> + x10<\/sup> + C
\n(c) (10x<\/sup> – x10<\/sup>) + C
\n(d) log (10x<\/sup> + x10<\/sup>) + C
\nSolution:
\n(d) log (10x<\/sup> + x10<\/sup>) + C
\n\"NCERT<\/p>\n

Question 39.
\n\\(\\int { \\frac { dx }{ { sin }^{ 2 }x{ \\quad cos }^{ 2 }x } = } \\)
\n(a) tan x + cot x + c
\n(b) tan x – cot x + c
\n(c) tan x cot x + c
\n(d) tan x – cot 2 x + c
\nSolution:
\n(b) tan x – cot x + c
\n\"NCERT<\/p>\n","protected":false},"excerpt":{"rendered":"

These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 Questions and Answers are prepared by our highly skilled subject experts. NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.2 Question 1. Solution: Let 1 + x\u00b2 = t \u21d2 2xdx = dt \u2234 \u222b dx = \u222b = log t …<\/p>\n

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2<\/span> Read More »<\/a><\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[3],"tags":[],"yoast_head":"\nNCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 - NCERT Solutions<\/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:\/\/ncertsolutions.guru\/ncert-solutions-for-class-12-maths-chapter-7-ex-7-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 - NCERT Solutions\" \/>\n<meta property=\"og:description\" content=\"These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 Questions and Answers are prepared by our highly skilled subject experts. 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NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.2 Question 1. Solution: Let 1 + x\u00b2 = t \u21d2 2xdx = dt \u2234 \u222b dx = \u222b = log t … NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 Read More »","og_url":"https:\/\/ncertsolutions.guru\/ncert-solutions-for-class-12-maths-chapter-7-ex-7-2\/","og_site_name":"NCERT Solutions","article_publisher":"https:\/\/www.facebook.com\/NCERTSolutionsGuru\/","article_published_time":"2024-02-08T12:11:44+00:00","article_modified_time":"2024-02-09T12:11:09+00:00","og_image":[{"url":"https:\/\/ncertsolutions.guru\/wp-content\/uploads\/2021\/01\/NCERT-Solutions-Guru.png"}],"twitter_card":"summary_large_image","twitter_creator":"@ncertsolguru","twitter_site":"@ncertsolguru","twitter_misc":{"Written by":"Prasanna","Est. reading time":"13 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/ncertsolutions.guru\/#organization","name":"NCERT Solutions","url":"https:\/\/ncertsolutions.guru\/","sameAs":["https:\/\/www.facebook.com\/NCERTSolutionsGuru\/","https:\/\/www.instagram.com\/ncertsolutionsguru\/","https:\/\/www.linkedin.com\/in\/ncertsolutionsguru\/","https:\/\/in.pinterest.com\/ncertsolutionsguru\/","https:\/\/en.wikipedia.org\/wiki\/National_Council_of_Educational_Research_and_Training","https:\/\/twitter.com\/ncertsolguru"],"logo":{"@type":"ImageObject","@id":"https:\/\/ncertsolutions.guru\/#logo","inLanguage":"en-US","url":"https:\/\/i1.wp.com\/ncertsolutions.guru\/wp-content\/uploads\/2020\/05\/NCERT-Solutions.png?fit=460%2C45&ssl=1","contentUrl":"https:\/\/i1.wp.com\/ncertsolutions.guru\/wp-content\/uploads\/2020\/05\/NCERT-Solutions.png?fit=460%2C45&ssl=1","width":460,"height":45,"caption":"NCERT Solutions"},"image":{"@id":"https:\/\/ncertsolutions.guru\/#logo"}},{"@type":"WebSite","@id":"https:\/\/ncertsolutions.guru\/#website","url":"https:\/\/ncertsolutions.guru\/","name":"NCERT Solutions","description":"NCERT Solutions for Class 1 to 12","publisher":{"@id":"https:\/\/ncertsolutions.guru\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/ncertsolutions.guru\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/ncertsolutions.guru\/ncert-solutions-for-class-12-maths-chapter-7-ex-7-2\/#primaryimage","inLanguage":"en-US","url":"https:\/\/i1.wp.com\/ncertsolutions.guru\/wp-content\/uploads\/2021\/01\/NCERT-Solutions-Guru.png?fit=170%2C17&ssl=1","contentUrl":"https:\/\/i1.wp.com\/ncertsolutions.guru\/wp-content\/uploads\/2021\/01\/NCERT-Solutions-Guru.png?fit=170%2C17&ssl=1","width":170,"height":17,"caption":"NCERT Solutions"},{"@type":"WebPage","@id":"https:\/\/ncertsolutions.guru\/ncert-solutions-for-class-12-maths-chapter-7-ex-7-2\/#webpage","url":"https:\/\/ncertsolutions.guru\/ncert-solutions-for-class-12-maths-chapter-7-ex-7-2\/","name":"NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 - 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