Integral of \( \displaystyle \frac{1}{6 x + \left(2 x - 1\right)^{2} - 3} \)

Problem 4.997 · medium

1. \[ \int \frac{1}{6 x + \left(2 x - 1\right)^{2} - 3}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{1}{4 x^{2} + 2 x - 2}\, dx \]
   algebra algebra algebraExpand the squared term. Combine like terms. Factor out the constant 2.✓ Proved
3. \[ = \int \frac{1}{\left(x + 1\right) \left(4 x - 2\right)}\, dx \]
   algebra algebraFactor the quadratic expression. Factor out 2 from the first linear term to simplify the fraction.✓ Proved
4. \[ = \int \left(- \frac{1}{6 x + 6} + \frac{1}{6 x - 3}\right)\, dx \]
   partial-fractionsDecompose the integrand using partial fractions.✓ Proved
5. \[ = \int \frac{1}{6 x - 3}\, dx - \int \frac{1}{6 x + 6}\, dx \]
   linearitySplit the integral into two parts.≈ Checked numerically
6. \[ = \frac{\ln{\left(x - \frac{1}{2} \right)}}{6} - \frac{\ln{\left(x + 1 \right)}}{6} \]
   antiderivativeEvaluate the integrals using the natural logarithm rule.≈ Checked numerically