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

Problem 4.279 · medium

1. \[ \int \left(2 x + \left(x - 1\right)^{3} - 2\right)\, dx \]
   integralStart with the integral of the given expression.✓ Proved
2. \[ = \int \left(x^{3} - 3 x^{2} + 5 x - 3\right)\, dx \]
   algebra simplifyExpand the binomial term (x - 1)**3. Combine like terms.✓ Proved
3. \[ = - \int 3\, dx + \int 5 x\, dx - \int 3 x^{2}\, dx + \int x^{3}\, dx \]
   linearitySplit the integral into individual terms.✓ Proved
4. \[ = \frac{x^{4}}{4} - x^{3} + \frac{5 x^{2}}{2} - 3 x \]
   antiderivativeIntegrate each term using the power rule.✓ Proved