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