# Integration by parts involving square root!

1. Mar 4, 2013

### learnonthefly

1. The problem statement, all variables and given/known data
|x3sqrt(4-x2)dx

2. Relevant equations
uv - | vdu

3. The attempt at a solution
u = x2 v = -1/3(4-x2)3/2
du = -2xdx dv = x(4-x2)1/2

uv - | vdu

x2(1/3)(4-x2)3/2 - | 1/3(4-x2)3/2(2xdx)
x2(1/3)(4-x2)3/2 +(1/3)|(4-x2)3/2(2xdx)

u = 4 - x2
du = -2xdx

x2(1/3)(4-x2)3/2 - (1/3)| u3/2du
x2(1/3)(4-x2)3/2-(1/3)2/5u5/2
x2(1/3)(4-x2)3/2-(1/3)(2/5)(4-x2)5/2
x2-(1/3)(4-x2)3/2-(2/15)(4-x2)5/2

Is this all I need? The answer I was supposed to get is -1/15(4-x2)3/2(8+3x2) I guess I dont see how that correlates

Last edited: Mar 4, 2013
2. Mar 4, 2013

### SammyS

Staff Emeritus
Hello learnonthefly. Welcome to PF !

I made a correction & did some editing above.

That is kind of hard to read.

What is the question that you have for us ?

3. Mar 4, 2013

### Curious3141

You didn't actually ask a question. And it's tough to read what you've written - please format in Latex, it makes things a lot easier.

At any rate, your final answer is wrong, but it's probably a simple sign error.

For example, in the first integration by parts, $v = -\frac{1}{3}{(4-x^2)}^{\frac{3}{2}}$. This is because of the $-x^2$ term within the parentheses.

You should go through your working carefully looking at all the sign errors and correct them.

EDIT: beaten by SammyS

4. Mar 4, 2013

### learnonthefly

Sorry guys, updated and cleaned

5. Mar 4, 2013

### SammyS

Staff Emeritus

$\displaystyle (1/15) (4-x^2)^{3/2} (3 x^2+8)$

You have a sign error in you integration by substitution part.

6. Mar 4, 2013

### learnonthefly

Yea I see where I messed up I fixed that above. Could the algebraically inclined show me the manipulation to get the answer?

7. Mar 4, 2013

### SammyS

Staff Emeritus
Factor $\displaystyle \ (4-x^2)^{3/2}\$ from your result & collect terms.

8. Mar 4, 2013

### iRaid

I think an easier way to do this would be make u=x^2 right from the beginning. Then after that make another substitution y=4-u. Then it's pretty easy from there.