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

Problem 4.154 · easy

1. \[ \int \ln{\left(2 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(\ln{\left(x \right)} + \ln{\left(2 \right)}\right)\, dx \]
   algebraUse the logarithm product rule to split the term.✓ Proved
3. \[ = \int \ln{\left(2 \right)}\, dx + \int \ln{\left(x \right)}\, dx \]
   linearitySplit the integral into two separate integrals.✓ Proved
4. \[ = x \ln{\left(2 \right)} + \int \ln{\left(x \right)}\, dx \]
   antiderivativeThe integral of a constant is the constant times x.✓ Proved
5. \[ = x \ln{\left(x \right)} + x \ln{\left(2 \right)} - \int 1\, dx \]
   partsUse integration by parts on the integral of log(x) with u=log(x) and dv=dx.✓ Proved
6. \[ = x \ln{\left(x \right)} - x + x \ln{\left(2 \right)} \]
   antiderivativeEvaluate the integral of 1 with respect to x.✓ Proved