∫_(3/2) ^2 (((x−1)/(3−x)))^(1/2) dx


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Question Number 628 by malwaan last updated on 14/Feb/15

$\int_{\frac{3}{2}}^{2} {(\frac{x - 1}{3 - x})}^{\frac{1}{2}} d x$
Commented by malwaan1 last updated on 26/Feb/15

$w h a t a b o u t t h e i n d e f i n i e i n t e g r a l ?$
	Answered by prakash jain last updated on 14/Feb/15

	$x = t^{2} + 1$ $d x = 2 t d t$ $\frac{x - 1}{3 - x} = \frac{t^{2}}{3 - (1 + t^{2})} = \frac{t^{2}}{2 - t^{2}}$ $\int \sqrt{\frac{x - 1}{3 - x}} = \int \frac{2 t^{2} d t}{\sqrt{2 - t^{2}}}$ $= \int \frac{2 t^{2} - 4}{\sqrt{2 - t^{2}}} d t + \int \frac{4}{\sqrt{2 - t^{2}}} d t$ $= - 2 \int \sqrt{2 - t^{2}} d t + \int \frac{4}{\sqrt{2 - t^{2}}} d t$ $= - 2 [\frac{1}{2} t \sqrt{2 - t^{2}} + \sin^{- 1} \frac{t}{\sqrt{2}}] + {4 \sin}^{- 1} \frac{t}{\sqrt{2}}$ $= {2 \sin}^{- 1} \frac{t}{\sqrt{2}} - t \sqrt{2 - t^{2}}$ $= {2 \sin}^{- 1} \sqrt{\frac{x - 1}{2}} - \sqrt{x - 1} \sqrt{3 - x}$ ${[{2 \sin}^{- 1} \sqrt{\frac{x - 1}{2}} - \sqrt{x - 1} \sqrt{3 - x}]}_{3 / 2}^{2}$ $= [{2 \sin}^{- 1} \sqrt{\frac{1}{2}} - 1] - [{2 \sin}^{- 1} \sqrt{\frac{1}{4}} - \sqrt{\frac{1}{2}} \sqrt{\frac{3}{2}}]$ $= 2 \frac{π}{4} - 1 - 2 \frac{π}{6} + \frac{\sqrt{3}}{2}$ $= \frac{π}{2} - \frac{π}{3} + \frac{\sqrt{3} - 2}{2}$ $= \frac{π}{6} + \frac{\sqrt{3} - 2}{2}$
	Commented by malwaan last updated on 03/Mar/15

	$p l e a s e s o l v e t h e i n t e g r a l b y s u b . x = 2 - c o s θ$ $w h a y t h e r e s u l t i s d i f f e r e n t ?$
	Commented by prakash jain last updated on 03/Mar/15

	$I reposted a question - may$ $be you can write what you got in that approach .$
	Commented by malwaan last updated on 03/Mar/15

	$I a m s o r r y$ $x = 2 - c o s θ$
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