Integral of \( \displaystyle \frac{\ln{\left(2 x \right)}}{4 x^{2}} \)

Problem 4.930 · medium

1. \[ \int \frac{\ln{\left(2 x \right)}}{4 x^{2}}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{\ln{\left(x \right)} + \ln{\left(2 \right)}}{4 x^{2}}\, dx \]
   rewriteUse the logarithm property log(ab) = log(a) + log(b).✓ Proved
3. \[ = \int \frac{\ln{\left(2 \right)}}{4 x^{2}}\, dx + \int \frac{\ln{\left(x \right)}}{4 x^{2}}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
4. \[ = \frac{\ln{\left(2 \right)} \int \frac{1}{x^{2}}\, dx}{4} + \int \frac{\ln{\left(x \right)}}{4 x^{2}}\, dx \]
   linearityFactor out the constants.✓ Proved
5. \[ = \int \frac{\ln{\left(x \right)}}{4 x^{2}}\, dx - \frac{\ln{\left(2 \right)}}{4 x} \]
   antiderivativeIntegrate the first term.✓ Proved
6. \[ = - \int \left(- \frac{1}{4 x^{2}}\right)\, dx - \frac{\ln{\left(x \right)}}{4 x} - \frac{\ln{\left(2 \right)}}{4 x} \]
   parts substitutionUse integration by parts on the second integral with u = log(x) and dv = x**(-2)/4 dx. Apply the derivative of log(x).✓ Proved
7. \[ = \int \frac{1}{4 x^{2}}\, dx - \frac{\ln{\left(x \right)}}{4 x} - \frac{\ln{\left(2 \right)}}{4 x} \]
   algebraDistribute the negative sign.✓ Proved
8. \[ = - \frac{\ln{\left(x \right)}}{4 x} - \frac{1}{4 x} - \frac{\ln{\left(2 \right)}}{4 x} \]
   antiderivative simplifyIntegrate the remaining term. Combine all terms into a single expression.✓ Proved