Example 27 - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 27 (Method 1) Evaluate ∫_0^1▒tan^(−1)𝑥/(1 + 𝑥^2 ) 𝑑𝑥 Step 1 : Let F(𝑥)=∫1▒tan^(−1)𝑥/(1+〖 𝑥〗^2 ) 𝑑𝑥 Put tan^(−1)𝑥=𝑡 Differentiating w.r.t.𝑥 𝑑/𝑑𝑥 (tan^(−1)𝑥 )=𝑑𝑡/𝑑𝑥 1/(1 + 𝑥^2 )=𝑑𝑡/𝑑𝑥 Therefore, ∫1▒tan^(−1)𝑥/(1+〖 𝑥〗^2 ) 𝑑𝑥=∫1▒〖𝑡/(1+𝑥^2 ) × (1+𝑥^2 )𝑑𝑡〗 =∫1▒〖𝑡 𝑑𝑡〗 =𝑡^2/2 Putting 𝑡=〖𝑡𝑎𝑛〗^(−1)𝑥 =(tan^(−1)𝑥 )^2/2 Hence 𝐹(𝑥)=(tan^(−1)𝑥 )^2/2 Step 2 : ∫1▒〖𝑡𝑎𝑛〗^(−1)𝑥/(1 + 𝑥^2 )=𝐹(1)−F(0) =1/2 (tan^(−1)1 )^2 −1/2 (tan^(−1)0 )^2 =1/2 (𝜋/4)^2−1/2 (0)^2 =1/2 𝜋^2/16 = 𝝅^𝟐/𝟑𝟐 Example 27 (Method 2) Evaluate ∫_0^1▒tan^(−1)𝑥/(1 + 𝑥^2 ) 𝑑𝑥 Put 𝑡=tan^(−1)𝑥 Differentiating w.r.t.𝑥 𝑑𝑡/𝑑𝑥=𝑑/𝑑𝑥 (tan^(−1)𝑥 ) 𝑑𝑡/𝑑𝑥=1/(1 + 𝑥^2 ) (1+𝑥^2 )𝑑𝑡=𝑑𝑥 Hence when value of x varies from 0 to 1, value of t varies from 0 to 𝜋/4 Therefore, ∫_0^1▒tan^(−1)𝑥/(1 + 𝑥^2 )=∫_0^(𝜋/4)▒𝑡/(1 + 𝑥^2 ) 𝑑𝑥 (1+𝑥^2 )𝑑𝑡 =∫_0^(𝜋/4)▒〖 𝑡 𝑑𝑡〗 =[𝑡^2/2]_0^(𝜋/4) =1/2 [(𝜋/4)^2−(0)^2 ] =1/2 × 𝜋^2/16 = 𝝅^𝟐/𝟑𝟐
Examples
Example 1 (ii)
Example 1 (iii)
Example 2 (i)
Example 2 (ii)
Example 2 (iii) Important
Example 3 (i)
Example 3 (ii) Important
Example 3 (iii)
Example 4
Example 5 (i)
Example 5 (ii)
Example 5 (iii) Important
Example 5 (iv) Important
Example 6 (i)
Example 6 (ii) Important
Example 6 (iii) Important
Example 7 (i)
Example 7 (ii) Important
Example 7 (iii)
Example 8 (i)
Example 8 (ii) Important
Example 9 (i)
Example 9 (ii) Important
Example 9 (iii) Important
Example 10 (i)
Example 10 (ii) Important
Example 11
Example 12
Example 13 Important
Example 14
Example 15 Important
Example 16 Important
Example 17
Example 18 Important
Example 19
Example 20 Important
Example 21 Important
Example 22 Important
Example 23
Example 24
Example 25 (i)
Example 25 (ii) Important
Example 25 (iii)
Example 25 (iv) Important
Example 26
Example 27 You are here
Example 28 Important
Example 29
Example 30
Example 31
Example 32 Important
Example 33 Important
Example 34 Important
Example 35
Example 36 Important
Example 37 Important
Example 38 Important
Example 39 Important
Example 40 Important
Example 41 Important
Example 42 Important
Question 1 Important
Question 2
Question 3 (Supplementary NCERT) Important
About the Author
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 and Computer Science at Teachoo