Integral of \( \displaystyle x^{2} e^{x} \)

Problem 4.144 · easy

1. \[ \int x^{2} e^{x}\, dx \]
   integralStart with the integral of the integrand.✓ Proved
2. \[ = x^{2} e^{x} - \int 2 x e^{x}\, dx \]
   partsApply integration by parts with u = x**2 and dv = exp(x) dx.✓ Proved
3. \[ = x^{2} e^{x} - 2 \int x e^{x}\, dx \]
   linearityPull the constant factor out of the integral.✓ Proved
4. \[ = x^{2} e^{x} - 2 x e^{x} + 2 \int e^{x}\, dx \]
   partsApply integration by parts to the remaining integral with u = x and dv = exp(x) dx.✓ Proved
5. \[ = x^{2} e^{x} - 2 x e^{x} + 2 e^{x} \]
   antiderivative algebraEvaluate the integral of exp(x). Distribute the constant and simplify.✓ Proved