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

Problem 4.732 · medium

1. \[ \int 2 x \ln{\left(2 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int 2 x \left(\ln{\left(x \right)} + \ln{\left(2 \right)}\right)\, dx \]
   algebraUse the logarithm product rule to expand the integrand.✓ Proved
3. \[ = 2 \ln{\left(2 \right)} \int x\, dx + 2 \int x \ln{\left(x \right)}\, dx \]
   linearityDistribute the 2*x and split the integral into two parts.✓ Proved
4. \[ = x^{2} \ln{\left(2 \right)} + 2 \int x \ln{\left(x \right)}\, dx \]
   antiderivativeIntegrate the first term.✓ Proved
5. \[ = x^{2} \ln{\left(x \right)} + x^{2} \ln{\left(2 \right)} - 2 \int \frac{x}{2}\, dx \]
   parts algebra algebraApply integration by parts to the second integral with u=log(x) and dv=x dx. Simplify the integrand inside the integral. Distribute the 2 into the parentheses.✓ Proved
6. \[ = x^{2} \ln{\left(x \right)} - \frac{x^{2}}{2} + x^{2} \ln{\left(2 \right)} \]
   antiderivative simplifyIntegrate the remaining term. Simplify the final expression.✓ Proved