Integral of \( \displaystyle \frac{\left(3 x - 1\right)^{2} + 1}{\left(3 x - 2\right) \left(3 x - 1\right)} \)

Problem 4.800 · hard

1. \[ \int \frac{\left(3 x - 1\right)^{2} + 1}{\left(3 x - 2\right) \left(3 x - 1\right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{9 x^{2} - 6 x + 2}{\left(3 x - 2\right) \left(3 x - 1\right)}\, dx \]
   algebraExpand the numerator.✓ Proved
3. \[ = \int \frac{9 x^{2} - 6 x + 2}{9 x^{2} - 9 x + 2}\, dx \]
   algebraExpand the denominator.✓ Proved
4. \[ = \int \left(\frac{3 x}{9 x^{2} - 9 x + 2} + 1\right)\, dx \]
   long-divisionPerform polynomial long division.✓ Proved
5. \[ = \int 1\, dx + \int \frac{3 x}{9 x^{2} - 9 x + 2}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
6. \[ = \int 1\, dx + \int \left(- \frac{1}{3 x - 1} + \frac{2}{3 x - 2}\right)\, dx \]
   partial-fractionsUse partial fraction decomposition on the remainder.✓ Proved
7. \[ = \int 1\, dx + \int \frac{2}{3 x - 2}\, dx + \int \left(- \frac{1}{3 x - 1}\right)\, dx \]
   linearitySplit the integral into three parts.≈ Checked numerically
8. \[ = x + \frac{2 \ln{\left(3 x - 2 \right)}}{3} - \frac{\ln{\left(3 x - 1 \right)}}{3} \]
   antiderivativeIntegrate each term separately.✓ Proved