Misc 35 - Chapter 7 Class 12 Integrals

Last updated at Dec. 8, 2016 by Teachoo

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

Transcript

Misc 35 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 Using ILATE, we take First function :- 𝑓(𝑥)=𝑥 Second function :- g(𝑥)=𝑒^𝑥 ∫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 = 𝑥 log⁡𝑥−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 )− (− 1) = 1 = R.H.S Hence, proved.

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.