Integral of \( \displaystyle 3 x \ln{\left(3 x \right)} \)

Problem 4.468 · medium

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