Ex 7.6, 3 - Chapter 7 Class 12 Integrals

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