Exercise
$\int\frac{x-3}{\left(x^2+1\right)\left(x-2\right)}dx$
Step-by-step Solution
Learn how to solve problems step by step online. Find the integral int((x-3)/((x^2+1)(x-2)))dx. Rewrite the fraction \frac{x-3}{\left(x^2+1\right)\left(x-2\right)} in 2 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{\frac{1}{5}x+\frac{7}{5}}{x^2+1}+\frac{-1}{5\left(x-2\right)}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{\frac{1}{5}x+\frac{7}{5}}{x^2+1}dx results in: \frac{1}{10}\ln\left(x^2+1\right)+\frac{7}{5}\arctan\left(x\right). Gather the results of all integrals.
Find the integral int((x-3)/((x^2+1)(x-2)))dx
Final answer to the exercise
$\frac{7}{5}\arctan\left(x\right)+\frac{1}{10}\ln\left|x^2+1\right|-\frac{1}{5}\ln\left|x-2\right|+C_0$