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

Problem 4.147 · medium

1. \[ \int \left(x + 1\right)^{2} e^{x + 1}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(x^{2} + 2 x + 1\right) e^{x + 1}\, dx \]
   algebraExpand the squared term.✓ Proved
3. \[ = \int 2 x e^{x + 1}\, dx + \int x^{2} e^{x + 1}\, dx + \int e^{x + 1}\, dx \]
   linearitySplit the integral into three parts.✓ Proved
4. \[ = e^{x + 1} + \int 2 x e^{x + 1}\, dx + \int x^{2} e^{x + 1}\, dx \]
   antiderivativeIntegrate the constant term.✓ Proved
5. \[ = 2 x e^{x + 1} + e^{x + 1} + \int x^{2} e^{x + 1}\, dx - \int 2 e^{x + 1}\, dx \]
   partsApply integration by parts to the second term.✓ Proved
6. \[ = 2 x e^{x + 1} - e^{x + 1} + \int x^{2} e^{x + 1}\, dx \]
   antiderivative simplifyIntegrate the remaining exponential term. Combine like terms.✓ Proved
7. \[ = x^{2} e^{x + 1} + 2 x e^{x + 1} - e^{x + 1} - \int 2 x e^{x + 1}\, dx \]
   partsApply integration by parts to the first term.✓ Proved
8. \[ = x^{2} e^{x + 1} - e^{x + 1} + \int 2 e^{x + 1}\, dx \]
   partsApply integration by parts to the integral term.✓ Proved
9. \[ = x^{2} e^{x + 1} + e^{x + 1} \]
   antiderivative simplifyIntegrate the remaining exponential term. Combine all terms.✓ Proved