Misc 21 - Chapter 7 Class 12 Integrals

Transcript

Misc 21 Integrate the function (𝑥^2 + 𝑥 + 1)/((𝑥 + 1)^2 (𝑥 + 2) ) ∫1▒〖(𝑥^2 + 𝑥 + 1)/((𝑥 + 1)^2 (𝑥 + 2) ) " " 𝑑𝑥〗 By partial fraction (𝑥^2 + 𝑥 + 1)/((𝑥 + 1)^2 (𝑥 + 2) )=A/(𝑥 + 2)+B/(𝑥 + 1)+C/〖(𝑥 + 1)〗^2 (𝑥^2 + 𝑥 + 1)/((𝑥 + 1)^2 (𝑥 + 2) )=(A〖(𝑥 + 1)〗^2 + B(𝑥 + 1)(𝑥 + 2) + C(𝑥 + 2))/(〖(𝑥 + 1)〗^2 (𝑥 + 2) ) Cancelling denominators 𝑥^2+𝑥+1=A〖 (𝑥+1)〗^2+B(𝑥+2)(𝑥+1)+C(𝑥+2) Hence, (𝑥^2 + 𝑥 + 1)/((𝑥 + 1)^2 (𝑥 + 2))=3/(𝑥 +2)−2/(𝑥 +1)+1/〖(𝑥 + 1)〗^2 ∫1▒(𝑥^2+ 𝑥 +1)/(〖(𝑥 + 1)〗^2 (𝑥 + 2))=∫1▒(3 𝑑𝑥)/(𝑥 + 2)−∫1▒(2 𝑑𝑥)/(𝑥 + 1)+∫1▒(1 𝑑𝑥)/(𝑥 + 1)^2 = 3log |𝑥+2| "– 2log " |𝑥+1|− 1/(𝑥 + 1)+𝐶 = "– 2log " |𝒙+𝟏|− 𝟏/(𝒙 + 𝟏)+"3log " |𝒙+𝟐|+𝑪