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\(\displaystyle I=\int_0^1 x^2e^x\,dx\)

Using IBP, let:

\(\displaystyle u=x^2\,\therefore\,du=2x\,dx\)

\(\displaystyle dv=e^x\,\therefore\,v=e^x\)

Hence:

\(\displaystyle I=\left[x^2e^x \right]_0^1-2\int_0^1\,xe^x\,dx=e-2\int_0^1\,xe^x\,dx\)

Using IBP again, let:

\(\displaystyle u=x\,\therefore\,du=dx\)

\(\displaystyle dv=e^x\,\therefore\,v=e^x\)

Hence:

\(\displaystyle I=e-2\left(\left[xe^x \right]_0^1-\int_0^1 e^x\,dx \right)=e-2\left(e-(e-1) \right)=e-2(1)=e-2\)