How do you integrate $int xe^(-x/2)$ by parts from $[0,4]$ ?

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Answer 1 · 2018-03-22T01:38:13

$-4(3e^-2-1)$

Let's select our values of $u, dv$ and integrate and differentiate for $du, v$ :

$u=x$
$du=dx$

$dv=e^(-x/2)dx$
$v=-2e^(-x/2)$

Note that our bounds of integration will not change -- we're not performing any substitutions.

So, for definite integrals, we integrate by parts in the following way:

$int_a^budv=uv|_a^b-int_a^bvdu$

Thus,

$int_0^4xe^(-x/2)dx=-2xe^(-x/2)|_0^4 +2int_0^4e^-x/2dx=(-8e^-2)-4e^-(x/2)|0^4=-8e^-2-4e^-2+4=-12e^-2+4=-4(3e^-2-1)$

How do you integrate int xe^(-x/2)int xe^(-x/2) by parts from [0,4][0,4]?