Prove definite integral 0->1 x ex dx = 1

Integral of e^(1/x)/x^2 (substitution)
Integral of e^(1/x)/x^2 (substitution)

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Misc 33 – Chapter 7 Class 12 Integrals

Last updated at June 13, 2023 by Teachoo

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Integration Formula Sheet – Chapter 41 Class 41 Formulas Important

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Last updated at June 13, 2023 by Teachoo

Misc 33 Prove that ∫_0^1▒𝑥 𝑒^𝑥 𝑑𝑥=1 Solving L.H.S ∫_0^1▒𝑥 𝑒^𝑥 𝑑𝑥 First we will solve ∫1▒𝒙 𝒆^𝒙 𝒅𝒙 ∫1▒𝑥 𝑒^𝑥 𝑑𝑥 ∫1▒𝑥 𝑒^𝑥 𝑑𝑥=𝑥∫1▒〖𝑒^𝑥 𝑑𝑥〗−∫1▒(𝑑𝑥/𝑑𝑥 ∫1▒〖𝑒^𝑥 𝑑𝑥〗) 𝑑𝑥 = 𝑥𝑒^𝑥−∫1▒1. 𝑒^𝑥 𝑑𝑥 = 𝑥𝑒^𝑥−𝑒^𝑥+𝐶 Applying limits ∫1_0^1▒〖𝑥 𝑒^𝑥 𝑑𝑥〗 = [𝑥𝑒^𝑥−𝑒^𝑥 ]_0^1 = (1𝑒^1−𝑒^1 )−(0.𝑒^0−𝑒^0) = (𝑒^1−𝑒^1 )− (0 − 1) = 1 = R.H.S Hence proved.

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