Evaluate the definite integral ${\int }_0^1{x}^2{e}^{x}dx$.

Solution

1. Apply integration by parts with $u=x^2$ and $dv=e^{x}dx$, giving $\int x^2{e}^{x}dx=x^2{e}^x-\int 2x{e}^{x}dx$.
2. Apply integration by parts again to $\int x{e}^{x}dx=x{e}^x-e^x$.
3. Combine the terms to get $$\begin{array}{rl}\int x^2{e}^{x}dx& =x^2{e}^x-2(x{e}^x-e^x)\\ & =e^x(x^2-2x+2)\end{array}$$.
4. Evaluate the expression at the limits 0 and 1: $[e^1(1-2+2)]-[e^0(0-0+2)]=e-2$.
5. Hence the answer is (A).