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

Problem 4.276 · medium

1. \[ \int \frac{2 x - 1}{2 x \left(2 x + 2\right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{2 x - 1}{4 x^{2} + 4 x}\, dx \]
   algebraExpand the denominator.✓ Proved
3. \[ = \int \left(\frac{2 x}{4 x^{2} + 4 x} - \frac{1}{4 x^{2} + 4 x}\right)\, dx \]
   linearitySplit the integrand into two parts.✓ Proved
4. \[ = \int \left(- \frac{1}{4 x^{2} + 4 x} + \frac{1}{2 x + 2}\right)\, dx \]
   simplifySimplify the first term by canceling x.✓ Proved
5. \[ = \int \frac{1}{2 x + 2}\, dx - \int \frac{1}{4 x^{2} + 4 x}\, dx \]
   linearitySplit the integral again.✓ Proved
6. \[ = - \int \frac{1}{4 x \left(x + 1\right)}\, dx + \int \frac{1}{2 x + 2}\, dx \]
   algebraFactor out the constant from the second denominator.✓ Proved
7. \[ = \int \frac{1}{2 x + 2}\, dx - \int \frac{1}{4 x^{2} + 4 x}\, dx \]
   algebraFactor out 4 from the second term.✓ Proved
8. \[ = - \frac{\int \frac{1}{x \left(x + 1\right)}\, dx}{4} + \int \frac{1}{2 x + 2}\, dx \]
   linearityPull out the constant factor.✓ Proved
9. \[ = \int \frac{1}{2 x + 2}\, dx - \frac{\int \left(- \frac{1}{x + 1} + \frac{1}{x}\right)\, dx}{4} \]
   partial-fractionsApply partial fraction decomposition to the second term.✓ Proved
10. \[ = - \frac{\int \frac{1}{x}\, dx}{4} + \frac{\int \frac{1}{x + 1}\, dx}{4} + \int \frac{1}{2 x + 2}\, dx \]
    linearityDistribute the constant and split the integral.✓ Proved
11. \[ = - \frac{\ln{\left(x \right)}}{4} + \frac{\ln{\left(x + 1 \right)}}{4} + \frac{\ln{\left(2 x + 2 \right)}}{2} \]
    antiderivativeEvaluate the integrals using the natural logarithm rule.✓ Proved