Integration By Parts?

Question

I am going through some of my recent homework assignments and I couldn't figure out the answer for this integration by parts problem.

Integral: x(e^3x)

Any help would be great!

Answer 1

u = x
du = dx

dv = e^3x
v = integral e^3x
v = (e^3x)/3

then rewrite, uv- integral vdu
so, (xe^3x)/3 - integral (e^3x)/3dx
therefore (1/9)(xe^3x) - ((e^3x)/3)

hope this helps

just remember uv - integral vdu

and always make you u such that du is simpler than u

Answer 2

Generally you want to choose for u the function that gets easier when you differentiate it. This is known as the ILATE rule. (see http://en.wikipedia.org/wiki/Integration_by_parts#The_ILATE_rule)
Here, e^3x never gets any easier, but the derivative of x is 1. Sodv is e^3x, v is (1/3) e^3x, and uv - vdu becomes
(1/3) x e^3x - integral (1/3) e^3x dx
This one is not so bad, sometimes you have to apply integeration by parts several times. In cases like that, tabular integraion aka the tic tac toe method comes in handy. Check the link I"ve given above, you'll see info on that as well.

Answer 3

let u = x
let dv = e^(3x)dx

good luck