# Use partial fractions to solve the integral int (x^2-x+20)/(x^3+5x)dx

Use partial fractions to solve the integral
\int \frac{x^2-x+20}{x^3+5x}dx
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$\int \frac{{x}^{2}-x+20}{{x}^{3}+5x}dx\phantom{\rule{0ex}{0ex}}\int \frac{{x}^{2}-x+20}{x\left({x}^{2}+5\right)}dx\phantom{\rule{0ex}{0ex}}\frac{A}{x}+\frac{Bx+C}{{x}^{2}+5}=\frac{{x}^{2}-x+20}{x\left({x}^{2}+5\right)}\phantom{\rule{0ex}{0ex}}A\left({x}^{2}+5\right)+Cx+5A={x}^{2}-x+20\phantom{\rule{0ex}{0ex}}A+B=1,C=-1,5A=20\phantom{\rule{0ex}{0ex}}B=-3,A=4\phantom{\rule{0ex}{0ex}}I=\int \left(\frac{4}{x}+\frac{-3x-1}{{x}^{2}+5}\right)dx\phantom{\rule{0ex}{0ex}}=4\mathrm{ln}x-\frac{3}{2}\int \frac{2xdx}{{x}^{2}+5}-\int \frac{dx}{{x}^{2}+5}\phantom{\rule{0ex}{0ex}}=4\mathrm{ln}x-\frac{3}{2}\mathrm{ln}\left({x}^{2}+5\right)-\frac{1}{\sqrt{5}}{\mathrm{tan}}^{\prime }\left(\frac{x}{\sqrt{5}}\right)+C$