Chapter 7 Class 12 Integrals
Concept wise

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Ex 7.9, 6 Evaluate the integrals using substitution ∫_0^(2 )▒𝑑𝑥/(𝑥 + 4 − 𝑥^2 ) We can write ∫_0^2▒〖𝑑𝑥/(𝑥 + 4 − 𝑥^2 )=∫_0^2▒𝑑𝑥/(−(𝑥^2 − 𝑥 − 4) )〗 =−∫_0^2▒𝑑𝑥/(𝑥^2 − 𝑥 − 4) =−∫_0^2▒𝑑𝑥/(𝑥^2 −2 × 1/2 × 𝑥 − 4) =−∫_0^2▒𝑑𝑥/(𝑥^2 −2 × 1/2 × 𝑥 + 1/2^2 − 1/2^2 − 4) =−∫_0^2▒𝑑𝑥/((𝑥 − 1/2)^2− 1/4 − 4) =−∫_0^2▒𝑑𝑥/((𝑥 − 1/2)^2− 17/4 ) =−∫_0^2▒𝑑𝑥/((𝑥 − 1/2)^2− (√17/4)^2 ) Let 𝑡=𝑥−1/2 Differentiating w.r.t.𝑥 𝑑𝑡/𝑑𝑥=1 𝑑𝑡=𝑑𝑥 When x varies from 0 to 2, then t varies from (−1)/2 to 3/2. Therefore, −∫_0^2▒〖𝑑𝑥/((𝑥 − 1/2)^2−(√17/2)^2 )=−∫_((−1)/2)^(3/2)▒𝑑𝑡/(𝑡 − (√17/2)^2 )〗 =−[1/2(√17/2) 𝑙𝑜𝑔|(𝑡 − √17/2)/(𝑡 + √17/2)|]_((−1)/( 2))^(3/2) =−1/√17 [𝑙𝑜𝑔|(3/2 − √17/2)/(3/2 + √17/2)|+𝑙𝑜𝑔|((−1)/( 2) − √17/2)/((−1)/( 2) + √17/2)|] =−1/√17 [𝑙𝑜𝑔|(3 − √17)/(3 + √17)|+𝑙𝑜𝑔|(−(1 + √17))/(−(1 − √17) )|] =−1/√17 𝑙𝑜𝑔|((3 − √17)/(3 + √17))/((1 + √17)/(1 − √17))| =−1/√17 𝑙𝑜𝑔|(3 − √17)/(3 + √17) ×(1 − √17)/(1 + √17)| =−1/√17 𝑙𝑜𝑔|(3+17 − 3√17 − √17)/(3 +17 + 3√17 + √17) | =−1/√17 𝑙𝑜𝑔|(20 − 4√17)/(20 + 4√17) | =−1/√17 𝑙𝑜𝑔|4(5 − √17)/4(5 + √17) | =−1/√17 𝑙𝑜𝑔|(5 − √17)/(5 + √17) | =1/√17 𝑙𝑜𝑔|(5 − √17)/(5 + √17) |^(−1) =1/√17 𝑙𝑜𝑔|(5 + √17)/(5 − √17)| =1/√17 𝑙𝑜𝑔|(5 + √17)/(5 − √17) ×(5 + √17)/(5 + √17)| =1/√17 𝑙𝑜𝑔|(5 − √17)^2/(5^2 − (√17)^2 ) | =1/√17 𝑙𝑜𝑔|(25 + 17 + 10√17)/(25 − 17) | =1/√17 𝑙𝑜𝑔|(42 + 10√17)/8 | =𝟏/√𝟏𝟕 𝒍𝒐𝒈|(𝟐𝟏 + 𝟓√𝟏𝟕)/𝟒 |

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.