Integral of \( \displaystyle 6 x + \left(3 x - 1\right)^{3} - 2 \)

Problem 4.874 · medium

1. \[ \int \left(6 x + \left(3 x - 1\right)^{3} - 2\right)\, dx \]
   integral algebraStart with the integral of the given expression. Rearrange the terms for clarity.✓ Proved
2. \[ = \int \left(3 x - 1\right)^{3}\, dx + \int \left(6 x - 2\right)\, dx \]
   linearitySplit the integral into two parts.✓ Proved
3. \[ = 3 x^{2} - 2 x + \int \left(3 x - 1\right)^{3}\, dx \]
   antiderivativeIntegrate the polynomial part.✓ Proved
4. \[ = 3 x^{2} - 2 x + \int \left(27 x^{3} - 27 x^{2} + 9 x - 1\right)\, dx \]
   algebraExpand the binomial term (3*x - 1)**3.✓ Proved
5. \[ = \frac{27 x^{4}}{4} - 9 x^{3} + \frac{15 x^{2}}{2} - 3 x \]
   antiderivative simplifyIntegrate the expanded polynomial term. Combine all terms and simplify the expression.✓ Proved