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- Integrate by partial fractions
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Rewrite the fraction $\frac{x^3+x^2+3}{\left(x^2+4\right)^2\left(x^2+x+1\right)}$ in $3$ simpler fractions using partial fraction decomposition
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$\frac{x-1}{\left(x^2+4\right)^2}+\frac{\frac{1}{13}x+\frac{4}{13}}{x^2+x+1}+\frac{-\frac{1}{13}x-\frac{3}{13}}{x^2+4}$
Learn how to solve problems step by step online. Find the integral int((x^3+x^2+3)/((x^2+4)^2(x^2+x+1)))dx. Rewrite the fraction \frac{x^3+x^2+3}{\left(x^2+4\right)^2\left(x^2+x+1\right)} in 3 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{x-1}{\left(x^2+4\right)^2}+\frac{\frac{1}{13}x+\frac{4}{13}}{x^2+x+1}+\frac{-\frac{1}{13}x-\frac{3}{13}}{x^2+4}\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{x-1}{\left(x^2+4\right)^2}dx results in: \frac{1}{-2\left(x^2+4\right)}-\frac{1}{16}\arctan\left(\frac{x}{2}\right)+\frac{-x}{8\left(x^2+4\right)^{\left(\frac{1}{2}+\frac{1}{2}\right)}}. Gather the results of all integrals.
