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Ex 7.6, 3 - Integrate x^2 e^x - Chapter 7 Class 12 NCERT

Ex 7.6, 3 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.6, 3 - Chapter 7 Class 12 Integrals - Part 3


Transcript

Ex 7.6, 3 Integrate the function 𝑥^2 𝑒𝑥 ∫1▒〖𝑥^2 𝑒^𝑥 𝑑𝑥〗 = 𝑥^2 ∫1▒〖𝑒𝑥 𝑑𝑥〗−∫1▒(𝑑(𝑥^2 )/𝑑𝑥 ∫1▒〖𝑒𝑥 𝑑𝑥〗) 𝑑𝑥 = 𝑥^2. 𝑒𝑥 −∫1▒〖2𝑥 . 𝑒𝑥〗 𝑑𝑥 = 𝑥^2. 𝑒𝑥 −2∫1▒〖𝒙 . 𝒆𝒙〗 𝒅𝒙 Now we know that ∫1▒〖𝑓(𝑥) 𝑔⁡(𝑥) 〗 𝑑𝑥=𝑓(𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥−∫1▒(𝑓′(𝑥)∫1▒𝑔(𝑥) 𝑑𝑥) 𝑑𝑥 Putting f(x) = x2 and g(x) = ex …(1) Solving I1 ∫1▒〖𝑥 𝑒^𝑥 𝑑𝑥〗 = 𝑥∫1▒𝑒𝑥 𝑑𝑥−∫1▒(𝑑𝑥/𝑑𝑥 ∫1▒𝑒^𝑥 𝑑𝑥) 𝑑𝑥 = 𝑥𝑒𝑥 −∫1▒𝑒𝑥 𝑑𝑥 = 𝑥𝑒𝑥 −𝑒𝑥 Now we know that ∫1▒〖𝑓(𝑥) 𝑔⁡(𝑥) 〗 𝑑𝑥=𝑓(𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥−∫1▒(𝑓′(𝑥)∫1▒𝑔(𝑥) 𝑑𝑥) 𝑑𝑥 Putting f(x) = x and g(x) = ex Putting value of I1 in our equation ∴ ∫1▒〖𝑥^2 𝑒𝑥" " 〗 𝑑𝑥" = " 𝑥^2. 𝑒𝑥 −2∫1▒〖𝒙 . 𝒆𝒙〗 𝒅𝒙 =𝑥^2. 𝑒𝑥 −2(𝒙𝒆𝒙−𝒆^𝒙 )+𝐶 =𝑥^2. 𝑒𝑥 −2𝑥𝑒𝑥+〖2𝑒〗^𝑥+𝐶 =𝒆𝒙 (𝒙^𝟐−𝟐𝒙+𝟐)+𝑪

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.