Integral of \( \displaystyle 4 x + \left(2 x + 1\right)^{3} + 2 \)

Problem 4.263 · medium

1. \[ \int \left(4 x + \left(2 x + 1\right)^{3} + 2\right)\, dx \]
   integralStart with the integral of the given expression.✓ Proved
2. \[ = \int 2\, dx + \int \left(4 x + \left(2 x + 1\right)^{3}\right)\, dx \]
   linearitySplit the integral into parts.✓ Proved
3. \[ = \int 2\, dx + \int 4 x\, dx + \int \left(2 x + 1\right)^{3}\, dx \]
   linearitySeparate the terms further.✓ Proved
4. \[ = \int 2\, dx + \int 4 x\, dx + \int \left(8 x^{3} + 12 x^{2} + 6 x + 1\right)\, dx \]
   algebraExpand the binomial (2x + 1)^3.✓ Proved
5. \[ = \int 1\, dx + \int 2\, dx + \int 4 x\, dx + \int 6 x\, dx + \int 12 x^{2}\, dx + \int 8 x^{3}\, dx \]
   linearityDistribute the integral over the expanded polynomial.✓ Proved
6. \[ = \int 3\, dx + \int 4 x\, dx + \int 6 x\, dx + \int 12 x^{2}\, dx + \int 8 x^{3}\, dx \]
   algebraCombine the constant terms 1 and 2.✓ Proved
7. \[ = 2 x^{4} + 4 x^{3} + 5 x^{2} + 3 x \]
   antiderivative simplify simplifyIntegrate each term individually. Simplify the coefficients. Combine like terms.✓ Proved