Final answer to the problem
$\frac{\left(x-1\right)^{2}\arctan\left(x\right)-2+x}{\left(x-1\right)^{2}}+C_0$
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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((2x+2)/((x^2+1)(x-1)^3))dx. Rewrite the fraction \frac{2x+2}{\left(x^2+1\right)\left(x-1\right)^3} in 4 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{1}{x^2+1}+\frac{2}{\left(x-1\right)^3}+\frac{-1}{\left(x-1\right)^{2}}\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{1}{x^2+1}dx results in: \arctan\left(x\right). The integral \int\frac{2}{\left(x-1\right)^3}dx results in: \frac{-1}{\left(x-1\right)^{2}}.
Final answer to the problem
$\frac{\left(x-1\right)^{2}\arctan\left(x\right)-2+x}{\left(x-1\right)^{2}}+C_0$
