Question

Integrate

Answer 1

This is an integration by parts problem, but fortunately there is a trick to reduce it a bit:

Answer 2

\[\int\limits x^2 e^{-x/4} dx = \int\limits (4u)^2 e^{-u} d(4u) = 4^3 \int\limits u^2 e^{-u} du \]

Answer 3

Where I used the substitution u = x / 4 so that x = 4u and dx = d(4u) = 4 du

Answer 4

Now let me just call everything y instead so I can use the u in the typical form of integration by parts:
\[\int\limits d(uv) = \int\limits u dv + \int\limits vdu\] Which if you drop the integral signs is merely the Liebniz rule for calculating derivatives:
\[\frac{ d }{ dx } (uv) = u \frac{ d }{ dx }v + v \frac{ d }{ dx }u\] Manipulating this integral term gives: \[ \int\limits u dv = uv - \int\limits v du\] So lets take: \[4^3 \int\limits y^2 e^{-y} dy\] where again I changed the name of the original subsitution variable to y=x/4 and get to work

Answer 5

I used integ by parts and I got