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Ex 7.6
Ex 7.6, 2 Important
Ex 7.6, 3 You are here
Ex 7.6, 4
Ex 7.6, 5 Important
Ex 7.6, 6
Ex 7.6, 7 Important
Ex 7.6, 8
Ex 7.6, 9
Ex 7.6, 10 Important
Ex 7.6, 11
Ex 7.6, 12
Ex 7.6, 13 Important
Ex 7.6, 14 Important
Ex 7.6, 15
Ex 7.6, 16
Ex 7.6, 17
Ex 7.6, 18 Important
Ex 7.6, 19
Ex 7.6, 20 Important
Ex 7.6, 21
Ex 7.6, 22 Important
Ex 7.6, 23 (MCQ)
Ex 7.6, 24 (MCQ) Important
Last updated at May 29, 2023 by Teachoo
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