Final answer to the problem
$\frac{1}{2}\ln\left|x^2+1\right|-\frac{1}{2}\arctan\left(x\right)+\frac{-x}{2\left(x^2+1\right)}+C_0$
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Step-by-step Solution
Learn how to solve integrals of rational functions problems step by step online. Find the integral int((x^3+x+-1)/((x^2+1)^2))dx. Rewrite the fraction \frac{x^3+x-1}{\left(x^2+1\right)^2} in 2 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{x}{x^2+1}+\frac{-1}{\left(x^2+1\right)^{2}}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{x}{x^2+1}dx results in: \frac{1}{2}\ln\left(x^2+1\right). The integral \int\frac{-1}{\left(x^2+1\right)^{2}}dx results in: -\left(\frac{1}{2}\arctan\left(x\right)+\frac{x}{2\left(x^2+1\right)}\right).
Final answer to the problem
$\frac{1}{2}\ln\left|x^2+1\right|-\frac{1}{2}\arctan\left(x\right)+\frac{-x}{2\left(x^2+1\right)}+C_0$
