# Integral :Complete the square and give a substitution (not necessarily trigonometric)

• Sep 30th 2010, 05:37 PM
highc1157
Integral :Complete the square and give a substitution (not necessarily trigonometric)
I have tried multiple substitutions and cannot get this homework problem right xD

$\displaystyle \int\dfrac{(x-1)}{\sqrt(2x-x^2)}dx$

everything in the denominator is under the square root
• Sep 30th 2010, 06:11 PM
skeeter
Quote:

Originally Posted by highc1157
I have tried multiple substitutions and cannot get this homework problem right xD

$\displaystyle \int\dfrac{(x-1)}{\sqrt(2x-x^2)}dx$

everything in the denominator is under the square root

$\displaystyle \displaystyle \int \frac{x-1}{\sqrt{1 -(x^2-2x+1)}} \, dx$

$\displaystyle \displaystyle \int \frac{x-1}{\sqrt{1 -(x-1)^2}} \, dx$

can you finish?
• Sep 30th 2010, 06:15 PM
sa-ri-ga-ma
After completing the square in the squareroot sing, the problem becomes

$\displaystyle \int\frac{1-x}{\sqrt{1 - (x-1)^2}}dx$

Substitue (x-1)^2 = t and proceed.
• Sep 30th 2010, 06:41 PM
highc1157
I did that substitution and got (x-2)(x-1), but not same as answer in the book.

I screwed up somewhere :(
• Sep 30th 2010, 06:49 PM
highc1157
hmmm the book answer is : $\displaystyle 1-(1-x)^2$

I tried it again, and got : $\displaystyle \sqrt{1-(1-x)^2$

I dont know where i messed up, simplified out everything and this is the integral that I take
$\displaystyle 1/2\int{(1-U)^(-(1/2))} du$ cant fix the exponent, supposed to be (1-U)^(-1/2)
• Sep 30th 2010, 06:59 PM
skeeter
$\displaystyle \displaystyle \int \frac{x-1}{\sqrt{1 -(x-1)^2}} \, dx$

let $\displaystyle t = (x-1)^2$

$\displaystyle dt = 2(x-1) \, dx$

$\displaystyle \displaystyle \frac{1}{2} \int \frac{2(x-1)}{\sqrt{1 - (x-1)^2}} \, dx$

$\displaystyle \displaystyle \frac{1}{2} \int \frac{dt}{\sqrt{1 - t}}$

$\displaystyle \displaystyle -\int \frac{-1}{2\sqrt{1 - t}} \, dt$

$\displaystyle \displaystyle -\sqrt{1-t} + C$

$\displaystyle \displaystyle -\sqrt{1-(x-1)^2} + C$

$\displaystyle \displaystyle -\sqrt{2x - x^2} + C$
• Sep 30th 2010, 07:01 PM
highc1157
that is not the answer in the book :( , i think i posted the book answer above
• Sep 30th 2010, 07:06 PM
skeeter
Quote:

Originally Posted by highc1157
that is not the answer in the book :( , i think i posted the book answer above

the "book" answer is incorrect ... they left off the radical and the constant of integration.

take the derivative of my solution ... what do you get?
• Sep 30th 2010, 07:10 PM
highc1157
Hey Skeeter,

Thanks for you're time and help you put into this , very much appreciated :)

But as for your answer and my answer they are slightly different, but this isn't right, what I got? $\displaystyle \sqrt{1-(1-x)^2$

oh wow you are right, The derivative of my solution does not equal the original equation !! grrrr lol
• Sep 30th 2010, 07:17 PM
skeeter
Quote:

Originally Posted by highc1157
Hey Skeeter,

Thanks for you're time and help you put into this , very much appreciated :)

But as for your answer and my answer they are slightly different, but this isn't right, what I got? $\displaystyle \sqrt{1-(1-x)^2$

oh wow you are right, The derivative of my solution does not equal the original equation !! grrrr lol

your solution is correct ... note that $\displaystyle \sqrt{1-(x-1)^2} = \sqrt{1 - (1-x)^2}$