Chapter 7 Class 12 Integrals
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Misc 28 - Chapter 7 Class 12 Integrals

Last updated at August 5, 2026 by

  Misc 28 - Definite integral dx / root 1+x - root x - Miscellaneous - Miscellaneous

part 2 - Misc 28 - Miscellaneous - Serial order wise - Chapter 7 Class 12 Integrals
part 3 - Misc 28 - Miscellaneous - Serial order wise - Chapter 7 Class 12 Integrals

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Misc 28 Evaluate the definite integral āˆ«_0^1ā–’怖š‘‘š‘„/(āˆš(1 + š‘„) āˆ’ āˆšš‘„) 怗 āˆ«_0^1ā–’怖š‘‘š‘„/(āˆš(1 + š‘„) āˆ’ āˆšš‘„) 怗 Rationalizing i.e., multiplying and dividing by (āˆš(1 + š‘„)+āˆšš‘„) = āˆ«_0^1ā–’怖š‘‘š‘„/(āˆš(1 + š‘„) āˆ’ āˆšš‘„) 怗Ɨ(āˆš(1 + š‘„) + āˆšš‘„)/(āˆš(1 + š‘„) + āˆšš‘„) . š‘‘š‘„ = āˆ«_0^1ā–’(āˆš(1 + š‘„) + āˆšš‘„)/((āˆš(1 + š‘„) )^2 āˆ’ (āˆšš‘„ )^2 ) . š‘‘š‘„ = āˆ«_0^1ā–’(āˆš(1 + š‘„) + āˆšš‘„)/(1 + š‘„ āˆ’ š‘„) . š‘‘š‘„ = āˆ«_0^1ā–’(āˆš(1 + š‘„) + āˆšš‘„)/1 . š‘‘š‘„ = āˆ«_0^1ā–’āˆš(1+š‘„) . š‘‘š‘„+āˆ«_0^1ā–’āˆšš‘„ . š‘‘š‘„" " = āˆ«_0^1ā–’(1+š‘„)^(1/2) š‘‘š‘„+āˆ«_0^1ā–’(š‘„)^(1/2) š‘‘š‘„" " = [(1 + š‘„)^(1/2 + 1)/(1/2 + 1)]_0^1 + [怖š‘„ 怗^(1/2 + 1)/(1/2 + 1)]_0^1 = [(1 + š‘„)^(3/2)/(3/2)]_0^1 + [怖š‘„ 怗^(3/2)/(3/2)]_0^1 = 怖2/3 [(1+š‘„)^(3/2) ]怗_0^1 + 2/3 [怖š‘„ 怗^(3/2) ]_0^1 = 2/3 [(1+1)^(3/2)āˆ’(1+0)^(3/2) ] + 2/3 [(1)^(3/2)āˆ’(0)^(3/2) ] = 2/3 [(2)^(3/2)āˆ’(1)^(3/2) ] + 2/3 [1āˆ’0] = 2/3 . (2)^(3/2)āˆ’2/3 [1]+2/3 [1] = 2/3 (2)^(3/2) = 2/3 [(2)^(1/2) ]^3 = 2/3 (āˆš2 )^3 = 2/3 . 2 āˆš2 = (šŸ’ āˆššŸ)/šŸ‘

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