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Misc 29 - Definite integral dx / root 1+x - root x - Miscellaneous

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Misc 29 Evaluate the definite integral ∫_0^1▒〖𝑑𝑥/(√(1 + 𝑥) − √𝑥) 〗 ∫_0^1▒〖𝑑𝑥/(√(1 + 𝑥) − √𝑥) 〗 Rationalizing i.e., multiplying and dividing by (√(1 + 𝑥)+√𝑥) = ∫_0^1▒〖𝑑𝑥/(√(1 + 𝑥) − √𝑥) 〗×(√(1 + 𝑥) + √𝑥)/(√(1 + 𝑥) + √𝑥) . 𝑑𝑥 = ∫_0^1▒(√(1 + 𝑥) + √𝑥)/((√(1 + 𝑥) )^2 − (√𝑥 )^2 ) . 𝑑𝑥 = ∫_0^1▒(√(1 + 𝑥) + √𝑥)/(1 + 𝑥 − 𝑥) . 𝑑𝑥 = ∫_0^1▒(√(1 + 𝑥) + √𝑥)/1 . 𝑑𝑥 = ∫_0^1▒√(1+𝑥) . 𝑑𝑥+∫_0^1▒√𝑥 . 𝑑𝑥" " = ∫_0^1▒(1+𝑥)^(1/2) 𝑑𝑥+∫_0^1▒(𝑥)^(1/2) 𝑑𝑥" " = [(1 + 𝑥)^(1/2 + 1)/(1/2 + 1)]_0^1 + [〖𝑥 〗^(1/2 + 1)/(1/2 + 1)]_0^1 = [(1 + 𝑥)^(3/2)/(3/2)]_0^1 + [〖𝑥 〗^(3/2)/(3/2)]_0^1 = 〖2/3 [(1+𝑥)^(3/2) ]〗_0^1 + 2/3 [〖𝑥 〗^(3/2) ]_0^1 = 2/3 [(1+1)^(3/2)−(1+0)^(3/2) ] + 2/3 [(1)^(3/2)−(0)^(3/2) ] = 2/3 [(2)^(3/2)−(1)^(3/2) ] + 2/3 [1−0] = 2/3 . (2)^(3/2)−2/3 [1]+2/3 [1] = 2/3 (2)^(3/2) = 2/3 [(2)^(1/2) ]^3 = 2/3 (√2 )^3 = 2/3 . 2 √2 = (𝟒 √𝟐)/𝟑

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.