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anonymous
 one year ago
Integrate
anonymous
 one year ago
Integrate

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0This is an integration by parts problem, but fortunately there is a trick to reduce it a bit:

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\int\limits x^2 e^{x/4} dx = \int\limits (4u)^2 e^{u} d(4u) = 4^3 \int\limits u^2 e^{u} du \]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Where I used the substitution u = x / 4 so that x = 4u and dx = d(4u) = 4 du

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Now 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

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I used integ by parts and I got

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1442185882375:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Wow that was wierd the page froze....

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\int\limits x^2 e^{x/4} dx = 4^3 \int\limits y^2 e^{y} dy\] where we choose u=y^2 and dv = e^y So: \[ \int\limits y^2 e^{y} dy = y^2 e^{y} + 2 \int\limits y e^{y} dy \]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Thank you so much =)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Now we choose u = y and dv = e^y (again) which gives: \[ \int\limits y e^{y} dy = ye^{y} + \int e^{y} dy = ye^{y}  e^{y} \] Putting It all together: \[ \int x^2 e^{x/4} dx = 4^3\int\limits y^2 e^{y} dy =4^3 ( y^2 e^{y} + 2 \int\limits y e^{y} dy ) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 4^3 ( y^2 e^{y} + 2 (ye^{y}  e^{y})) = 64e^{y}(y^2 2 y + 1) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 64e^{y}(y1)^2\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Now dont forget to substitute back in y=x/4 so your answer is in terms of the original function

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[ \int x^2 e^{x/4} dx = 64e^{x/4}( \frac{x}{4} 1)^2 \]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Well I messed up the exponent but you get the point

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Hey how did you get that human calculator tag btw?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0The smartscore thing..

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ahhh yea I am still relatively new here, but, as you can see, I do believe I am eligible for such a title :D :D

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I just need more medals

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Anyways see you around
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