Exercise
$\int\frac{x^2-2x}{x^3-x^2+x-1}dx$
Step-by-step Solution
Learn how to solve problems step by step online. Find the integral int((x^2-2x)/(x^3-x^2x+-1))dx. Rewrite the expression \frac{x^2-2x}{x^3-x^2+x-1} inside the integral in factored form. Rewrite the fraction \frac{x^2-2x}{\left(x^{2}+1\right)\left(x-1\right)} in 2 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{\frac{3}{2}x-\frac{1}{2}}{x^{2}+1}+\frac{-1}{2\left(x-1\right)}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{\frac{3}{2}x-\frac{1}{2}}{x^{2}+1}dx results in: \frac{3}{4}\ln\left(x^{2}+1\right)-\frac{1}{2}\arctan\left(x\right).
Find the integral int((x^2-2x)/(x^3-x^2x+-1))dx
Final answer to the exercise
$-\frac{1}{2}\arctan\left(x\right)+\frac{3}{4}\ln\left|x^{2}+1\right|-\frac{1}{2}\ln\left|x-1\right|+C_0$