Integral
∫[(x-1)^(-3/2) ][(2-x)^(-1/2)]dx
Let's try this again.
$\displaystyle \int\frac{1}{(x-1)^{\frac{3}{2}}\sqrt{2-x}}dx$
Let $\displaystyle u=\sqrt{2-x}, \;\ 2-u^{2}=x, \;\ -2udu=dx$
Make the subs and get:
$\displaystyle -2\int{(1-u^{2})^{\frac{-3}{2}}}du$
Let $\displaystyle u=sin{\theta}, \;\ du=cos{\theta}d{\theta}$
$\displaystyle -2\int{(1-sin^{2}{\theta})^{\frac{-3}{2}}cos{\theta}}d{\theta}$
$\displaystyle -2\int{sec^{2}{\theta}}d{\theta}$
$\displaystyle -2tan{\theta}$
Resub:
$\displaystyle -2tan(sin^{-1}u)$
$\displaystyle \frac{-2u}{\sqrt{1-u^{2}}}$
Resub:
$\displaystyle \boxed{\frac{-2\sqrt{2-x}}{\sqrt{x-1}}}$