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Misc 35 - Prove definite integral 0->1 x ex dx = 1 - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 35 Prove that ∫_0^1β–’π‘₯ 𝑒^π‘₯ 𝑑π‘₯=1 Solving L.H.S ∫_0^1β–’π‘₯ 𝑒^π‘₯ 𝑑π‘₯ First we will solve ∫1▒𝒙 𝒆^𝒙 𝒅𝒙 ∫1β–’π‘₯ 𝑒^π‘₯ 𝑑π‘₯ Using ILATE, we take First function :- 𝑓(π‘₯)=π‘₯ Second function :- g(π‘₯)=𝑒^π‘₯ ∫1β–’π‘₯ 𝑒^π‘₯ 𝑑π‘₯=π‘₯∫1▒〖𝑒^π‘₯ 𝑑π‘₯γ€—βˆ’βˆ«1β–’(𝑑π‘₯/𝑑π‘₯ ∫1▒〖𝑒^π‘₯ 𝑑π‘₯γ€—) 𝑑π‘₯ = π‘₯𝑒^π‘₯βˆ’βˆ«1β–’1. 𝑒^π‘₯ 𝑑π‘₯ = π‘₯𝑒^π‘₯βˆ’π‘’^π‘₯+𝐢 = π‘₯ log⁑π‘₯βˆ’π‘₯+𝐢 Applying limits ∫1_0^1β–’γ€–π‘₯ 𝑒^π‘₯ 𝑑π‘₯γ€— = [π‘₯𝑒^π‘₯βˆ’π‘’^π‘₯ ]_0^1 = (1𝑒^1βˆ’π‘’^1 )βˆ’(0.𝑒^0βˆ’π‘’^0) = (𝑒^1βˆ’π‘’^1 )βˆ’ (βˆ’ 1) = 1 = R.H.S Hence, proved.

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