Exercise
$\int\frac{2x^2-4x+5}{\left(3x+1\right)\left(3x-1\right)\left(2x+1\right)}dx$
Step-by-step Solution
Learn how to solve problems step by step online. Find the integral int((2x^2-4x+5)/((3x+1)(3x-1)(2x+1)))dx. Rewrite the expression \frac{2x^2-4x+5}{\left(3x+1\right)\left(3x-1\right)\left(2x+1\right)} inside the integral in factored form. Rewrite the fraction \frac{2x^2-4x+5}{\left(9x^2-1\right)\left(2x+1\right)} in 2 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{-26x+11}{9x^2-1}+\frac{6}{2x+1}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{-26x+11}{9x^2-1}dx results in: -\frac{13}{9}\ln\left(x^2-\frac{1}{9}\right)-\frac{11}{6}\ln\left(3x+1\right)+\frac{11}{6}\ln\left(3x-1\right).
Find the integral int((2x^2-4x+5)/((3x+1)(3x-1)(2x+1)))dx
Final answer to the exercise
$-\frac{11}{6}\ln\left|3x+1\right|+\frac{11}{6}\ln\left|3x-1\right|-\frac{13}{9}\ln\left|x^2-\frac{1}{9}\right|+3\ln\left|2x+1\right|+C_0$