Integral of \( \displaystyle \frac{6 x - 4}{- 2 x + \left(2 x - 1\right)^{2} + 1} \)

Problem 4.471 · hard

1. \[ \int \frac{6 x - 4}{- 2 x + \left(2 x - 1\right)^{2} + 1}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{6 x - 4}{4 x^{2} - 6 x + 2}\, dx \]
   algebra simplifyExpand the squared term. Combine like terms in the denominator.✓ Proved
3. \[ = \int \frac{3 x - 2}{2 x^{2} - 3 x + 1}\, dx \]
   algebraDivide the numerator and denominator by 2.✓ Proved
4. \[ = \int \frac{3 x - 2}{\left(x - 1\right) \left(2 x - 1\right)}\, dx \]
   algebraFactor the quadratic denominator.✓ Proved
5. \[ = \int \left(\frac{1}{2 x - 1} + \frac{1}{x - 1}\right)\, dx \]
   partial-fractionsPerform partial fraction decomposition.✓ Proved
6. \[ = \int \frac{1}{x - 1}\, dx + \int \frac{1}{2 x - 1}\, dx \]
   linearitySplit the integral into two parts.≈ Checked numerically
7. \[ = \ln{\left(x - 1 \right)} + \frac{\ln{\left(2 x - 1 \right)}}{2} \]
   antiderivativeIntegrate both terms using the substitution rule.✓ Proved