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

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- Sep 30th 2010, 05:37 PMhighc1157Integral :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 PMskeeter
- Sep 30th 2010, 06:15 PMsa-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 PMhighc1157
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 PMhighc1157
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 PMskeeter
$\displaystyle \displaystyle \int \frac{x-1}{\sqrt{1 -(x-1)^2}} \, dx$

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 PMhighc1157
that is not the answer in the book :( , i think i posted the book answer above

- Sep 30th 2010, 07:06 PMskeeter
- Sep 30th 2010, 07:10 PMhighc1157
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 PMskeeter