Integration of x^2 dx / (1-x)^1/2

[ganondorf29]

Homework Statement

Integrate: x^2 dx / (1-x)^1/2

Homework Equations

U substitution

The Attempt at a Solution

First I defined u = (1-x)^1/2
du = -dx/2(1-x)^1/2
dx = -2(1-x)^1/2 du

then for x
u^2 = 1-x
x = 1-u^2

integral of: (1-u^2)^2/u
= (u^4-2u^2+1) / u
= u^3-2u+1/u

integrating:

u^4/4 - u^2 + 1/u

= ((1-x)^2)/4 - (1-x) + ln|1-x| + CIs this right?

 

[naresh]

You can try differentiating your answer to check if it is correct..

 

[ganondorf29]

I differentiated and the answer I got was wrong. Does anyone know what I did wrong?

 

[Unassuming]

looks like you should have gotten ,

ln| (1-x)^1/2 |

EDIT: I was wrong here, look at Dick's hint later. It seems pretty easy after his hint.

 

[Dick]

You figured out that dx=-2u*du. But then you never used it. There's no logs in the problem at all.

 

[ganondorf29]

If dx = -2u*du that means that I would have to set u = (1-x)^1/2 and square both sides and then take the derivative, right? Well, if I do that, what do I do for the x^2 on the numerator?

 

[Dick]

If dx = -2u*du that means that I would have to set u = (1-x)^1/2 and square both sides and then take the derivative, right? Well, if I do that, what do I do for the x^2 on the numerator?

What?? No, you are doing everything right, but you forgot to replace the dx with an expression for du. You said "dx = -2(1-x)^1/2 du". Use that expression.

 

[ganondorf29]

Ok, I reworked the dx part and I got

Integral of:
((1-u^2)^2*-2u*du)/u
= -2(1-u^2)^2
= -2(u^4-2u^2+1)

= -(2/5)(u^5) + (4/3)(u^3) - 2u + C

= -(2/5)((1-x)^5) + (4/3)((1-x)^3) - 2(1-x) + C

But when I differentiate this I am not getting x^2/sqrt(1-x)

 

[Dick]

u=sqrt(1-x). You substituted u=(1-x).

 

[ganondorf29]

After fixing the substitution

-(2/5)((sqrt(1-x))^5) + (4/3)((sqrt(1-x))^3) - 2(sqrt(1-x)) + C

d/dx of -(2/5)((sqrt(1-x))^5) + (4/3)((sqrt(1-x))^3) - 2(sqrt(1-x)) + C

= 1/sqrt(1-x) - (x+1)*(sqrt(1-x))

which isn't x^2/(sqrt(1-x))


I went to mathematica's online integral calculator and they gave this:
-(2/15)*(sqrt(1-x)*(x(3x+4)+8

My Ti-89 gave the same answer once factored. I still don't know what I'm doing wrong.

 

[Dick]

1/sqrt(1-x) - (x+1)*(sqrt(1-x)). Multiply the numerator and denominator of the second term by sqrt(1-x). 1/sqrt(1-x)-(x+1)(1-x)/sqrt(1-x)=(1-(1-x^2))/sqrt(1-x)=x^2/sqrt(1-x). It's right. You've got too many machines telling you what to do.

 

[ganondorf29]

Thank you so much