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# $\int \frac{{x}^{3}+{x}^{2}+2x+1}{{x}^{2}-x+1}dx$

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Solution

## $\mathrm{Let}I=\int \left(\frac{{x}^{3}+{x}^{2}+2x+1}{{x}^{2}-x+1}\right)dx\phantom{\rule{0ex}{0ex}}{x}^{2}-x+1\stackrel{x+2}{\overline{){x}^{3}+{x}^{2}+2x+1}}\phantom{\rule{0ex}{0ex}}{x}^{3}-{x}^{2}+x\phantom{\rule{0ex}{0ex}}\overline{)-+-}\phantom{\rule{0ex}{0ex}}2{x}^{2}+x+1\phantom{\rule{0ex}{0ex}}2{x}^{2}-2x+2\phantom{\rule{0ex}{0ex}}\overline{)-+-}\phantom{\rule{0ex}{0ex}}3x-1\phantom{\rule{0ex}{0ex}}\mathrm{Therefore},\phantom{\rule{0ex}{0ex}}\frac{{x}^{3}+{x}^{2}+2x+1}{{x}^{2}-x+1}=x+2+\frac{3x-1}{{x}^{2}-x+1}.....\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{Let}\phantom{\rule{0ex}{0ex}}3x-1=A\frac{d}{dx}\left({x}^{2}-x+1\right)+B\phantom{\rule{0ex}{0ex}}3x-1=A\left(2x-1\right)+B\phantom{\rule{0ex}{0ex}}3x-1=\left(2A\right)x+B-A\phantom{\rule{0ex}{0ex}}\mathrm{Equating}\mathrm{Coefficients}\mathrm{of}\mathrm{like}\mathrm{terms}\phantom{\rule{0ex}{0ex}}2A=3\phantom{\rule{0ex}{0ex}}A=\frac{3}{2}\phantom{\rule{0ex}{0ex}}B-A=-1\phantom{\rule{0ex}{0ex}}B-\frac{3}{2}=-1\phantom{\rule{0ex}{0ex}}B=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\int \left(\frac{{x}^{3}+{x}^{2}+2x+1}{{x}^{2}-x+1}\right)dx=\int \left(x+2\right)dx+\int \left(\frac{\frac{3}{2}\left(2x-1\right)+\frac{1}{2}}{{x}^{2}-x+1}\right)dx\phantom{\rule{0ex}{0ex}}=\int \left(x+2\right)dx+\frac{3}{2}\int \left(\frac{2x-1}{{x}^{2}-x+1}\right)dx+\frac{1}{2}\int \frac{dx}{{x}^{2}-x+1}\phantom{\rule{0ex}{0ex}}=\int \left(x+2\right)dx+\frac{3}{2}\int \frac{\left(2x-1\right)dx}{{x}^{2}-x+1}+\frac{1}{2}\int \frac{dx}{{x}^{2}-x+\frac{1}{4}-\frac{1}{4}+1}\phantom{\rule{0ex}{0ex}}=\int \left(x+2\right)dx+\frac{3}{2}\int \frac{\left(2x-1\right)dx}{{x}^{2}-x+1}+\frac{1}{2}\int \frac{dx}{{\left(x-\frac{1}{2}\right)}^{2}+{\left(\frac{\sqrt{3}}{2}\right)}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{{x}^{2}}{2}+2x+\frac{3}{2}\mathrm{log}\left|{x}^{2}-x+1\right|+\frac{1}{2}×\frac{2}{\sqrt{3}}{\mathrm{tan}}^{-1}\left(\frac{x-\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)+C\phantom{\rule{0ex}{0ex}}=\frac{{x}^{2}}{2}+2x+\frac{3}{2}\mathrm{log}\left|{x}^{2}-x+1\right|+\frac{1}{\sqrt{3}}{\mathrm{tan}}^{-1}\left(\frac{2x-1}{\sqrt{3}}\right)+C$

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