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Question Number 43517 by Raj Singh last updated on 11/Sep/18

Commented by maxmathsup by imad last updated on 11/Sep/18

$$letI=\int \frac{x^3+1}{x^2-5x+6}dx\Rightarrow I=\int \frac{x(x^2-5x+6)+5x^2-6x+1}{x^2-5x+6}dx$$$$=\int xdx+\int \frac{5x^2-6x+1}{x^2-5x+6}dx$$$$=\frac{x^2}{2}+\int \frac{5(x^2-5x+6)+25x-29}{x^2-5x+6}dx$$$$=\frac{x^2}{2}+5x+\int \frac{25x-29}{x^2-5x+6}dxletdecomposeF\left(x\right)=\frac{25x-29}{x^2-5x+6}$$$$\mathrm{\Delta }=25-24=1\Rightarrow x_1=\frac{5+1}{2}=3and\mathrm{\Delta }=\frac{5-1}{2}=2\Rightarrow F\left(x\right)=\frac{25x-29}{(x-2)(x-3)}$$$$=\frac{a}{x-2}+\frac{b}{x-3}$$$$a={lim}_{x\to 2}(x-2)F\left(x\right)=\frac{50-29}{-1}=-21$$$$b={lim}_{x\to 3}(x-3)F\left(x\right)=75-29=46\Rightarrow F\left(x\right)=\frac{-21}{x-2}+\frac{46}{x-3}\Rightarrow$$$$I=\frac{x^2}{2}+5x-21ln\mid x-2\mid +46ln\mid x-3\mid +c.$$$$$$

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