Ex 7.2, 11 - Integrate x / root (x+4) - Class 12 NCERT - Ex 7.2

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise

Transcript

Ex 7.2, 11 (Method 1) Integrate the function: π‘₯/√(π‘₯ + 4) , π‘₯>0 π‘₯/√(π‘₯ + 4) , π‘₯>0 Step 1: Simplify the given function π‘₯/√(π‘₯ + 4) = (π‘₯ + 4 βˆ’ 4)/√(π‘₯ + 4) = (π‘₯ + 4)/√(π‘₯ + 4) βˆ’ 4/√(π‘₯ + 4) = (π‘₯ + 4)^(1/2 βˆ’ 1) βˆ’ 4(π‘₯ + 4)^(1/2) = (π‘₯ + 4)^(1/2) βˆ’ 4(π‘₯ + 4)^(βˆ’ 1/2) Step 2: Integrating the function ∫1β–’γ€–" " π‘₯/√(π‘₯ + 4)γ€— . 𝑑π‘₯ = ∫1β–’((π‘₯ + 4)^(1/2) " βˆ’ " γ€–4 (π‘₯ + 4)γ€—^(βˆ’ 1/2) ) . 𝑑π‘₯ = ∫1β–’(π‘₯ + 4)^(1/2) . 𝑑π‘₯ βˆ’ 4∫1β–’(π‘₯ + 4)^(βˆ’ 1/2) . 𝑑π‘₯ = (π‘₯ + 4)^(1/2 + 1)/(1/2 + 1) βˆ’ (4 (π‘₯ + 4)^(βˆ’ 1/2 + 1))/(βˆ’ 1/2 + 1) + C = (π‘₯ + 4)^(3/2)/(3/2 ) βˆ’ (4 (π‘₯ + 4)^(1/2))/( 1/2) + C = 2/3 (π‘₯+4)^(3/2) βˆ’ 4.2 (π‘₯+4)^(1/2) + C = γ€–2(π‘₯+4)γ€—^(1/2) ((π‘₯ + 4)/3 βˆ’4) + 𝐢 = γ€–2(π‘₯+4)γ€—^(1/2) ((π‘₯ + 4 βˆ’12)/3) + 𝐢 = γ€–2(π‘₯+4)γ€—^(1/2) ((π‘₯ βˆ’ 8))/3 + 𝐢 = 𝟐/πŸ‘ √(𝒙 + πŸ’) (π’™βˆ’πŸ–) + π‘ͺ Ex 7.2, 11 (Method 2) Integrate the function: π‘₯/√(π‘₯ + 4) , π‘₯>0 Step 1: Let π‘₯+4=𝑑 Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ 1+0= 𝑑𝑑/𝑑π‘₯ 1= 𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑 Step 2: Integrating the function ∫1β–’γ€–" " π‘₯/√(π‘₯ + 4)γ€— . 𝑑π‘₯ Putting 𝑑=π‘₯+4 & 𝑑π‘₯=𝑑𝑑 = ∫1β–’π‘₯/βˆšπ‘‘ . 𝑑𝑑 = ∫1β–’(𝑑 βˆ’ 4)/βˆšπ‘‘ . 𝑑𝑑 = ∫1β–’(𝑑 βˆ’ 4)/βˆšπ‘‘ . 𝑑𝑑 = ∫1β–’(𝑑/βˆšπ‘‘ βˆ’4/βˆšπ‘‘) 𝑑𝑑 = ∫1▒𝑑/βˆšπ‘‘ . 𝑑𝑑 βˆ’ ∫1β–’4/βˆšπ‘‘ . 𝑑𝑑 = ∫1▒𝑑^(1 βˆ’ 1/2) . 𝑑𝑑 βˆ’ ∫1β–’γ€–4 . 𝑑^(βˆ’ 1/2) γ€—. 𝑑𝑑 = ∫1▒𝑑^(1/2) . 𝑑𝑑 βˆ’ ∫1β–’γ€–4 . 𝑑^(βˆ’ 1/2) γ€—. 𝑑𝑑 = 𝑑^(1/2 + 1)/(1/2 + 1) βˆ’ 4 𝑑^(βˆ’ 1/2 + 1)/(βˆ’ 1/2 + 1) +𝐢 = (𝑑^(3/2) )/(3/2) βˆ’ 4 𝑑^(1/2)/(1/2) +𝐢 = 2/3 𝑑^(3/2) βˆ’ 4 . 2𝑑^(1/2) +𝐢 Taking 2/3 . 𝑑^(1/2) as common , we get = 2/3 . 𝑑^(1/2) (π‘‘βˆ’4 . 3)+𝐢 = 2/3 . 𝑑^(1/2) (π‘‘βˆ’12)+𝐢 Putting the value of 𝑑=π‘₯+4 = 2/3 . (π‘₯+4)^(1/2) (π‘₯+4βˆ’12)+𝐢 = 𝟐/πŸ‘ √(𝒙+πŸ’) (π’™βˆ’πŸ–)+π‘ͺ

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.