CBSE Class 12 Sample Paper for 2021 Boards

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Question 33 - CBSE Class 12 Sample Paper for 2021 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at April 16, 2024 by

Find ∫ x 2 + 1 / (x 2 + 2) (x 2 + 3) dx

Find Integration ∫ (x^2 + 1)/ (x^2 + 2) (x^2 + 3) dx - Teachoo Maths

Question 33 - CBSE Class 12 Sample Paper for 2021 Boards - Part 2
Question 33 - CBSE Class 12 Sample Paper for 2021 Boards - Part 3

Transcript

Question 33 Find ∫1▒〖(𝑥^2 + 1)/((𝑥^2 + 2) (𝑥^2 + 3)) 𝑑𝑥〗 Putting 𝒙^𝟐=𝒚 (𝑥^2 + 1 )/((𝑥^2 + 2) (𝑥^2 + 3) )=(𝑦 + 1)/((𝑦 + 2) (𝑦 + 3) ) We can write this in form (𝑦 + 1)/((𝑦 + 2) (𝑦 + 3) )=𝐴/((𝑦 + 2) ) + 𝐵/((𝑦 + 3) ) (𝑦 + 1)/((𝑦 + 2) (𝑦 + 3) )=(𝐴(𝑦 +3) + 𝐵 (𝑦 + 2))/((𝑦 + 2) (𝑦 + 3) ) By cancelling denominator 𝑦+1=𝐴(𝑦 +3) + 𝐵 (𝑦 + 2) Putting y = −3 −3+1=𝐴(−3+3)+𝐵(−3+2) −2=𝐴 × 0+𝐵 × −1 −2=−𝐵 𝑩=𝟐 Putting y = −2 −2+1=𝐴(−2+3)+𝐵(−2+2) −1=𝐴 × 1+𝐵 × 0 −1=𝐴 𝑨=−𝟏 Hence we can write (𝑦 + 1)/((𝑦 + 2) (𝑦 + 3) )=(−1)/((𝑦 + 2) ) + 2/((𝑦 + 3) ) Substituting back 𝑦=𝑥^2 (𝑥^2 + 1 )/((𝑥^2 + 2) (𝑥^2 + 3) ) =(−1)/((𝑥^2 + 2) )+2/((𝑥^2 + 3) ) Therefore, ∫1▒(𝑥^2 + 1 )/((𝑥^2 + 2) (𝑥^2 + 3) ) 𝑑𝑥=∫1▒(−1)/((𝑥^2 + 2) ) 𝑑𝑥+∫1▒2/((𝑥^2 + 3) ) 𝑑𝑥 =−∫1▒1/((𝑥^2 +(√2)^2 ) ) 𝑑𝑥+2∫1▒1/((𝑥^2 +(√3)^2 ) ) 𝑑𝑥 By using formula ∫1▒1/(𝑥^2 + 𝑎^2 ) 𝑑𝑥=1/𝑎 〖𝑡𝑎𝑛〗^(−1)⁡(𝑥/𝑎)+𝐶 =(−𝟏)/√𝟐 〖𝒕𝒂𝒏〗^(−𝟏)⁡〖𝒙/√𝟐〗+𝟐/√𝟑 〖𝒕𝒂𝒏〗^(−𝟏)⁡〖𝒙/√𝟑〗 +𝑪

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.