Misc 33 - Chapter 7 Class 12 Integrals

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Misc 33 - Prove definite integral 0->1 x ex dx = 1 - Miscellaneous - Miscellaneous

part 2 - Misc 33 - Miscellaneous - Serial order wise - Chapter 7 Class 12 Integrals

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Misc 33 Prove that āˆ«_0^1ā–’š‘„ š‘’^š‘„ š‘‘š‘„=1 Solving L.H.S āˆ«_0^1ā–’š‘„ š‘’^š‘„ š‘‘š‘„ First we will solve āˆ«1ā–’š’™ š’†^š’™ š’…š’™ āˆ«1ā–’š‘„ š‘’^š‘„ š‘‘š‘„ āˆ«1ā–’š‘„ š‘’^š‘„ š‘‘š‘„=š‘„āˆ«1ā–’怖š‘’^š‘„ š‘‘š‘„怗āˆ’āˆ«1ā–’(š‘‘š‘„/š‘‘š‘„ āˆ«1ā–’怖š‘’^š‘„ š‘‘š‘„怗) š‘‘š‘„ = š‘„š‘’^š‘„āˆ’āˆ«1ā–’1. š‘’^š‘„ š‘‘š‘„ = š‘„š‘’^š‘„āˆ’š‘’^š‘„+š¶ Applying limits āˆ«1_0^1ā–’怖š‘„ š‘’^š‘„ š‘‘š‘„怗 = [š‘„š‘’^š‘„āˆ’š‘’^š‘„ ]_0^1 = (1š‘’^1āˆ’š‘’^1 )āˆ’(0.š‘’^0āˆ’š‘’^0) = (š‘’^1āˆ’š‘’^1 )āˆ’ (0 āˆ’ 1) = 1 = R.H.S Hence proved.

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