Integral of \( \displaystyle 3 x - 1 + \frac{1}{3 x - 1} \)

Problem 4.552 · medium

1. \[ \int \left(3 x - 1 + \frac{1}{3 x - 1}\right)\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = - \int 1\, dx + \int 3 x\, dx + \int \frac{1}{3 x - 1}\, dx \]
   linearitySplit the integral into three parts.✓ Proved
3. \[ = - \int 1\, dx + \int 3 x\, dx + \int \frac{1}{3 \left(x - \frac{1}{3}\right)}\, dx \]
   algebraRewrite the fraction using algebraic manipulation.✓ Proved
4. \[ = - \int 1\, dx + \int 3 x\, dx + \frac{\int \frac{1}{x - \frac{1}{3}}\, dx}{3} \]
   linearityPull the constant factor out of the integral.✓ Proved
5. \[ = \frac{3 x^{2}}{2} - x + \frac{\ln{\left(3 x - 1 \right)}}{3} \]
   antiderivativeCompute the antiderivative of each term.✓ Proved