calculate-dx-x-1-2-x-

Question Number 63721 by mathmax by abdo last updated on 08/Jul/19

c a l c u l a t e \int \frac{d x}{\sqrt{(x - 1) (2 - x)}}

Commented by Prithwish sen last updated on 08/Jul/19

put x - l = z^{2} \Rightarrow dx = 2 zdz

\int \frac{2 zdz}{z \sqrt{3 - z^{2}}} = 2 \int \frac{dz}{\sqrt{(\sqrt{{3)}^{2}} - z^{2}}}

Commented by mathmax by abdo last updated on 08/Jul/19

l e t I = \int \frac{d x}{\sqrt{(x - 1) (2 - x)}} w e h a v e (x - 1) (2 - x) = 2 x - x^{2} - 2 + x

- x^{2} + 3 x - 2 = - (x^{2} - 3 x + 2) = - (x^{2} - 2 \frac{3}{2} x + \frac{9}{4} + 2 - \frac{9}{4})

= - {{(x - \frac{3}{2})}^{2} - \frac{1}{4}} = \frac{1}{4} - {(x - \frac{3}{2})}^{2} f o r t h a t w e u s e t h e c h a n g e m e n t

x - \frac{3}{2} = \frac{s i n t}{2} \Rightarrow I = \int \frac{c o s t d t}{2 \frac{1}{2} \sqrt{1 - {s i n}^{2} t}} = \int \frac{c o s t}{c o s t} d t + c

= \int d t + c = t + c = a r c s i n (2 x - 3) + c .

Answered by MJS last updated on 08/Jul/19

\int \frac{d x}{\sqrt{(x - 1) (2 - x)}} =

[t = \arccos (2 x - 3) \to d x = - \sqrt{(x - 1) (2 - x)} d t]

= - \int d t = - t = - \arccos (2 x - 3) + C

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