find the antiderivative of x^2e^(3x)

Limits, differentiation, related rates, integration, trig integrals, etc.
yabo2k
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find the antiderivative of x^2e^(3x)

find the antiderivative of x^2e^(3x)

Martingale
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Re: find the antiderivative

find the antiderivative of x^2e^(3x)
use integration by parts

yabo2k
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Re: find the antiderivative of x^2e^(3x)

i get ((x^2e^3x)/3)-((2xe^3x)/9)-((2e^3x)/9)+C

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Re: find the antiderivative of x^2e^(3x)

i get ((x^2e^3x)/3)-((2xe^3x)/9)-((2e^3x)/9)+C
close...

look at the last bit...(2e^3x)/9

yabo2k
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Re: find the antiderivative of x^2e^(3x)

would it be e^(3x)/3?

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Re: find the antiderivative of x^2e^(3x)

would it be e^(3x)/3?
no

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Re: find the antiderivative of x^2e^(3x)

find the antiderivative of x^2e^(3x)

$\int x^2e^{3x}dx$

Spoiler:
$u=x^2$ $dv=e^{3x}dx$
$du=2x dx$ $v=\frac{e^{3x}}{3}$

$\int x^2e^{3x}dx=x^2\frac{e^{3x}}{3}-\int 2x\frac{e^{3x}}{3}dx$

$u=2x$ $dv=\frac{e^{3x}}{3}dx$
$du=2 dx$ $v=\frac{e^{3x}}{9}$

$=x^2\frac{e^{3x}}{3}-\int 2x\frac{e^{3x}}{3}dx=x^2\frac{e^{3x}}{3}-2x\frac{e^{3x}}{9}+\int 2\frac{e^{3x}}{9}dx$

$=x^2\frac{e^{3x}}{3}-2x\frac{e^{3x}}{9}+2\frac{e^{3x}}{27}+c$

$=\left(\frac{x^2}{3}-\frac{2x}{9}+\frac{2}{27}\right)e^{3x}+c$

yabo2k
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Re: find the antiderivative of x^2e^(3x)

I'll do it step by step

udv=(x^2)(e^3x/3)-(e^3x/3)(2xdx)

=(x^2e^3x)/3-(2xe^3x)dx/3)

Then do I need to find the antiderivative of the second part (2xe^3x)dx/3) because there are two fnc (2x/3) and (e^3x)

u= 2x/3 dv= (e^3x)dx
du= (2/3)dx v= (e^3x/3)

am I going in the right direction?

Martingale
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Re: find the antiderivative of x^2e^(3x)

I'll do it step by step

udv=(x^2)(e^3x/3)-(e^3x/3)(2xdx)

=(x^2e^3x)/3-(2xe^3x)dx/3)

Then do I need to find the antiderivative of the second part (2xe^3x)dx/3) because there are two fnc (2x/3) and (e^3x)

u= 2x/3 dv= (e^3x)dx
du= (2/3)dx v= (e^3x/3)

am I going in the right direction?
yes

yabo2k
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Re: find the antiderivative of x^2e^(3x)

for the antiderivative of the second part then I would get

(2x/3)(e^3x/3)-(e^3x/3)((2/3)dx)