Question

Calcule a integral indefinida, use a substituição por partes:
$\int\ x^{2}. e^{x} \, dx$

Answer 1

$\boxed{\boxed{\int udv=uv-\int vdu}}$

Entre potências de x e exponenciais, escolhemos a potência de x como u
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$u=x^{2}~~~~~~~\rightarrow~~~~~~~du=2xdx\\v=e^{x}~~~~~~~\leftarrow~~~~~~~dv=e^{x}dx$

Então:

$\int x^{2}e^{x}dx=uv-\int vdu=x^{2}e^{x}-\int e^{x}\cdot 2xdx=x^{2}e^{x}-2\int xe^{x}dx$
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Vamos integrar xe^x por partes:

$u=x~~~~~~~\rightarrow~~~~~~~du=dx\\v=e^{x}~~~~~~\leftarrow~~~~~~~dv=e^{x}dx\\\\\int e^{x}dx=xe^{x}-\int e^{x}dx=xe^{x}-e^{x}=e^{x}(x-1)$
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Substituindo:

$\int x^{2}e^{x}dx=x^{2}e^{x}-2\int xe^{x}dx\\\\\int x^{2}e^{x}dx=x^{2}e^{x}-2e^{x}(x-1)+C\\\\\int x^{2}e^{x}dx=x^{2}e^{x}-e^{x}(2x-2)+C\\\\\boxed{\boxed{\int x^{2}e^{x}dx=e^{x}(x^{2}-2x+2)+C}}$