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

Problem 4.926 · hard

1. \[ \int \frac{9 x + 2}{- 3 x + \left(3 x + 1\right)^{2} - 1}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{9 x + 2}{9 x^{2} + 3 x}\, dx \]
   algebra algebraExpand the squared term. Combine like terms in the denominator.✓ Proved
3. \[ = \int \frac{9 x + 2}{3 x \left(3 x + 1\right)}\, dx \]
   algebraFactor the denominator.✓ Proved
4. \[ = \int \frac{3 + \frac{2}{3 x}}{3 x + 1}\, dx \]
   algebraDivide the numerator by the common factor 3.✓ Proved
5. \[ = \int \frac{3 x + \frac{2}{3}}{x \left(3 x + 1\right)}\, dx \]
   algebraDistribute the division to simplify the fraction.✓ Proved
6. \[ = \int \left(\frac{1}{3 x + 1} + \frac{2}{3 x}\right)\, dx \]
   partial-fractionsDecompose the integrand using partial fractions.✓ Proved
7. \[ = \int \frac{2}{3 x}\, dx + \int \frac{1}{3 x + 1}\, dx \]
   linearitySplit the integral into two parts.≈ Checked numerically
8. \[ = \frac{2 \ln{\left(x \right)}}{3} + \frac{\ln{\left(3 x + 1 \right)}}{3} \]
   antiderivativeIntegrate each term separately.✓ Proved