Primitive of x squared by Exponential of a x/Examples/x squared by e^-x

Theorem

$\ds \int x^2 e^{-x} \rd x = -e^{-x} \paren {x^2 + 2 x + 2} + C$

Proof

From Primitive of $x^2 e^{a x}$:

$\ds \int x^2 e^{a x} \rd x = \frac {e^{a x} } a \paren {x^2 - \frac {2 x} a + \frac 2 {a^2} } + C$

So:

\(\ds \int x^2 e^{-x} \rd x\)

\(=\)

\(\ds \frac {e^{-x} } {-1} \paren {x^2 - \frac {2 x} {-1} + \frac 2 {\paren {-1}^2} } + C\)

Primitive of $x^2 e^{a x}$: setting $a = -1$

\(\ds \)

\(=\)

\(\ds -e^{-x} \paren {x^2 + 2 x + 2} + C\)

$\blacksquare$