I need SERIOUS HELP integration by parts

I need SERIOUS HELP...integration by parts!!!

Homework Statement

Integrate: x^3*e^x

The Attempt at a Solution

I have the answer in my book but I am not understanding why you have to repeat the integration 3 times...

1.) dv = e^x dx
v = e^x
u = x^3
du = 3x^2

S=integral sign lol
S u*dv = uv - S v*du
S x^3*e^3 = x^3*e^x - S 3x^2*e^x dx

And now I have to do this 2 more times but I don't understand why and how I am supposed to know to do that?

$$I=\int x^3 e^xdx$$

$$u=x^3$$
$$du=3x^2dx$$

$$dV=e^xdx$$
$$V=e^x$$

$$I=x^3e^x-3\int x^2e^xdx$$

$$u=x^2$$
$$du=2xdx$$

$$dV=e^xdx$$
$$V=e^x$$

$$I=x^3e^x-3\left(x^2-2\int xe^xdx\right)$$

Now do it again.

But why? How do I know that I need to do this 2, 3, 4 times or whatever it may be?

Well to me, just keep going till you've reduced it to where it is no longer a product, since Integration by Parts is the reverse of the product rule.

$$\int xe^xdx$$ = product

$$\int xdx$$ = not a product

But why? How do I know that I need to do this 2, 3, 4 times or whatever it may be?

if it has an x^n * sin, cos, e^x, or any of those terms that repeat when integrated you will need to do integration by parts n times

for this problem you'll have:

x^3
3x^2
6x
1

you'll just keep repeating until the x^n term becomes 1.

if it has an x^n * sin, cos, e^x, or any of those terms that repeat when integrated you will need to do integration by parts n times

for this problem you'll have:

x^3
3x^2
6x
1

you'll just keep repeating until the x^n term becomes 1.
Yep, and those are called recurrent formulas, or recurrent integration.

tiny-tim
Homework Helper
… the lesser evil …

But why? How do I know that I need to do this 2, 3, 4 times or whatever it may be?

Hi BuBbLeS01! e^x is nice. We like e^x. It behaves itself.

x^3 is bad. We want to get rid of it.

So we wave our magic wand and make it smaller.

Then again. Then again, as many times as are necessary to make it disappear.

(They teach an incantation as well, at Hogwarts - but it's not strictly necessary.)

It's a tiresome job … but somebody has to do it …

It's just a good-versus-evil thing! 