# I need SERIOUS HELP integration by parts

1. Mar 18, 2008

### BuBbLeS01

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

1. The problem statement, all variables and given/known data
Integrate: x^3*e^x

2. Relevant equations

3. 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?

2. Mar 18, 2008

### rocomath

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

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

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

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

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

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

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

Now do it again.

3. Mar 18, 2008

### BuBbLeS01

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

4. Mar 18, 2008

### rocomath

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

5. Mar 18, 2008

### bob1182006

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.

6. Mar 18, 2008

### sutupidmath

Yep, and those are called recurrent formulas, or recurrent integration.

7. Mar 19, 2008

### tiny-tim

… the lesser evil …

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!