Integral of \( \displaystyle \frac{9 x^{2} + 1}{3 x \left(3 x - 1\right)} \)

Problem 4.460 · medium

1. \[ \int \frac{9 x^{2} + 1}{3 x \left(3 x - 1\right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{9 x^{2} + 1}{9 x^{2} - 3 x}\, dx \]
   algebra algebraExpand the denominator. Rewrite the numerator to facilitate long division.✓ Proved
3. \[ = \int \left(\frac{3 x + 1}{9 x^{2} - 3 x} + 1\right)\, dx \]
   long-divisionPerform polynomial long division.✓ Proved
4. \[ = \int 1\, dx + \int \frac{3 x + 1}{3 x \left(3 x - 1\right)}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
5. \[ = \int 1\, dx + \int \left(\frac{2}{3 x - 1} - \frac{1}{3 x}\right)\, dx \]
   partial-fractionsApply partial fraction decomposition to the second term.✓ Proved
6. \[ = \int 1\, dx - \int \frac{1}{3 x}\, dx + \int \frac{2}{3 x - 1}\, dx \]
   linearitySplit the integral into three separate terms.≈ Checked numerically
7. \[ = x - \frac{\ln{\left(x \right)}}{3} + \frac{2 \ln{\left(3 x - 1 \right)}}{3} \]
   antiderivativeCompute the antiderivative of each term.✓ Proved