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Question Number 20242 by tammi last updated on 24/Aug/17

∫((5x^2 +11x+26)/(x^2 +2x+5))dx  integration by partial fraction

$$\int\frac{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{11}{x}+\mathrm{26}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}}{dx} \\ $$$${integration}\:{by}\:{partial}\:{fraction} \\ $$

Answered by ajfour last updated on 24/Aug/17

((5x^2 +11x+26)/(x^2 +2x+5))=A+((Bx+C)/(x^2 +2x+5))  ⇒A(x^2 +2x+5)+Bx+C                   =5x^2 +11x+26  ⇒ A=5,  2A+B=11,  5A+C=26  ⇒B=1, C=1  I=5∫dx+∫((x+1)/(x^2 +2x+1))dx    =5x+(1/2)ln (x^2 +2x+1)+c .

$$\frac{\mathrm{5}{x}^{\mathrm{2}} +\mathrm{11}{x}+\mathrm{26}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}}={A}+\frac{{Bx}+{C}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}} \\ $$$$\Rightarrow{A}\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}\right)+{Bx}+{C} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{5}{x}^{\mathrm{2}} +\mathrm{11}{x}+\mathrm{26} \\ $$$$\Rightarrow\:{A}=\mathrm{5},\:\:\mathrm{2}{A}+{B}=\mathrm{11},\:\:\mathrm{5}{A}+{C}=\mathrm{26} \\ $$$$\Rightarrow{B}=\mathrm{1},\:{C}=\mathrm{1} \\ $$$${I}=\mathrm{5}\int{dx}+\int\frac{{x}+\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}{dx} \\ $$$$\:\:=\mathrm{5}{x}+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}\right)+{c}\:. \\ $$

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