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Find ∫1β–’γ€–(x + 1)/((x^2Β  + 1)Β  x) dxγ€—

This question is similar to Ex 13.2, 2 - Chapter 13 Class 12 - Probability

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Transcript

Question 7 Find ∫1β–’γ€–(π‘₯ + 1)/((π‘₯^2 + 1) π‘₯) 𝑑π‘₯γ€—Let (π‘₯ + 1)/((π‘₯^2 + 1) π‘₯) = (𝑨𝒙 + 𝑩)/((𝒙^𝟐 + 𝟏) ) + π‘ͺ/𝒙 (π‘₯ + 1)/((π‘₯^2 + 1) π‘₯) = ((𝐴π‘₯ + 𝐡)π‘₯ + 𝐢(1 + π‘₯^2 ))/((π‘₯^2 + 1) π‘₯) By cancelling denominator 𝒙 + 𝟏 = (𝑨𝒙 + 𝑩)𝒙 + π‘ͺ(𝟏 + 𝒙^𝟐 ) Putting 𝒙=𝟎 0 + 1 = (𝐴(0) + 𝐡) Γ— 0 + 𝐢(1 +0^2 ) 1 = 𝐢 π‘ͺ = 𝟏 Putting 𝒙=𝟏 1 + 1 = (𝐴(1) + 𝐡) Γ— 1 + 𝐢(1 +1^2 ) 2 = (𝐴 + 𝐡) +2𝐢 Putting 𝐢 = 1 2 = (𝐴 + 𝐡) +2 Γ— 1 2 = (𝐴 + 𝐡) +2 2βˆ’2 = (𝐴 + 𝐡) 0 = 𝐴 + 𝐡 𝑨=βˆ’ 𝑩 Putting 𝒙=βˆ’πŸ βˆ’1 + 1 = (𝐴(βˆ’1) + 𝐡) Γ— βˆ’1 + 𝐢(1 +γ€–(βˆ’1)γ€—^2 ) 0 = βˆ’(βˆ’π΄ + 𝐡) +𝐢 Γ— (1+1) 0 = βˆ’(βˆ’π΄ + 𝐡) +2𝐢 Putting 𝐴=βˆ’ 𝐡 0 = βˆ’(𝐡 + 𝐡) +2𝐢 0 = βˆ’2B +2𝐢 2B =2𝐢 B =𝐢 Putting 𝐢 = 1 𝑩 = 𝟏 And, 𝐴=βˆ’π΅ ∴ 𝑨=βˆ’πŸ Thus, 𝐴=βˆ’1, 𝐡=1, 𝐢 = 1 So, we can write (𝒙 + 𝟏)/((𝒙^𝟐 + 𝟏) 𝒙) = (𝐴π‘₯ + 𝐡)/((π‘₯^2 + 1) ) + 𝐢/π‘₯ = ((βˆ’1)π‘₯ +1)/((π‘₯^2 + 1) ) + 1/π‘₯ = (βˆ’π’™ + 𝟏)/((𝒙^𝟐 + 𝟏) ) + 𝟏/𝒙 Therefore integrating ∫1β–’(π‘₯ + 1)/((π‘₯^2 + 1) π‘₯) 𝑑π‘₯ = ∫1β–’(βˆ’π’™ + 𝟏)/((𝒙^𝟐 + 𝟏) ) 𝑑π‘₯ + ∫1β–’1/(π‘₯ ) 𝑑π‘₯ = ∫1β–’(βˆ’π‘₯ + 1)/((π‘₯^2 + 1) ) 𝑑π‘₯ + ∫1β–’1/(π‘₯ ) 𝑑π‘₯ = ∫1β–’(βˆ’π‘₯)/((π‘₯^2 + 1) ) 𝑑π‘₯ + ∫1β–’1/(π‘₯^2 + 1) 𝑑π‘₯ + ∫1β–’1/(π‘₯ ) 𝑑π‘₯ = ∫1β–’(βˆ’π‘₯)/((π‘₯^2 + 1) ) 𝑑π‘₯ + γ€–π­πšπ§γ€—^(βˆ’πŸ)⁑𝒙 + π’π’π’ˆβ‘γ€–|𝒙|γ€—+𝐢 = ∫1β–’(βˆ’π‘₯)/((π‘₯^2 + 1) ) 𝑑π‘₯ + γ€–π­πšπ§γ€—^(βˆ’πŸ)⁑𝒙 + π’π’π’ˆβ‘γ€–|𝒙|γ€—+𝐢 Solving 𝐈1 I1 = ∫1β–’(βˆ’π‘₯)/(π‘₯^2 + 1) 𝑑π‘₯ Let 𝒕 = 𝒙^𝟐+𝟏 𝑑𝑑/𝑑π‘₯ = 2π‘₯ 𝑑𝑑/2π‘₯ = 𝑑π‘₯ Hence ∫1β–’(βˆ’π‘₯)/(π‘₯^2 + 1) 𝑑π‘₯ = ∫1β–’γ€–(βˆ’π‘₯)/𝑑 . 𝑑𝑑/2π‘₯γ€— = βˆ’βˆ«1▒𝑑𝑑/2(𝑑) = (βˆ’1)/2 γ€–log 〗⁑|𝑑|+𝐢1 Putting back t = π‘₯^2+1 = (βˆ’πŸ)/𝟐 γ€–π’π’π’ˆ 〗⁑|𝒙^𝟐+𝟏|+π‘ͺ𝟐 Therefore integrating ∫1β–’(π‘₯ + 1)/((π‘₯^2 + 1) π‘₯) 𝑑π‘₯ = (βˆ’πŸ)/𝟐 γ€–π’π’π’ˆ 〗⁑|𝒙^𝟐+𝟏| + γ€–π­πšπ§γ€—^(βˆ’πŸ)⁑𝒙 + π’π’π’ˆβ‘γ€–|𝒙|γ€—+𝐢

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.