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

Problem 4.795 · hard

1. \[ \int \left(\frac{2 x + 1}{2 x} + \frac{1}{2 x \left(2 x + 1\right)}\right)\, dx \]
   algebra simplifySplit the fraction using the numerator's terms. Cancel common terms in the first part of the integrand.✓ Proved
2. \[ = \int \left(1 + \frac{1}{2 x} + \frac{1}{2 x \left(2 x + 1\right)}\right)\, dx \]
   algebraDivide the numerator by the denominator in the first term.✓ Proved
3. \[ = \int 1\, dx + \int \frac{1}{2 x}\, dx + \int \frac{1}{2 x \left(2 x + 1\right)}\, dx \]
   linearityDistribute the integral across the sum.✓ Proved
4. \[ = \int 1\, dx + \int \frac{1}{2 x}\, dx + \int \left(- \frac{1}{2 x + 1} + \frac{1}{2 x}\right)\, dx \]
   rewriteUse partial fraction decomposition on the third term.✓ Proved
5. \[ = \int 1\, dx + 2 \int \frac{1}{2 x}\, dx - \int \frac{1}{2 x + 1}\, dx \]
   linearity simplifySplit the integral of the decomposed term. Combine the identical integral terms.≈ Checked numerically
6. \[ = x + \ln{\left(2 x \right)} - \frac{\ln{\left(2 x + 1 \right)}}{2} \]
   antiderivative simplifyIntegrate each term separately. Simplify the coefficients.✓ Proved
7. \[ = x + \ln{\left(2 x \right)} - \ln{\left(\sqrt{2 x + 1} \right)} \]
   rewriteRewrite the division by 2 as a square root inside the logarithm.✓ Proved
8. \[ = x + \ln{\left(\frac{2 x}{\sqrt{2 x + 1}} \right)} \]
   simplifyUse logarithmic properties to combine the terms into a single logarithm.≈ Checked numerically