dx-x-a-b-x-

Question Number 122697 by john santu last updated on 19/Nov/20

\int \frac{d x}{\sqrt{(x - a) (b - x)}} ?

Answered by liberty last updated on 19/Nov/20

S o l v e φ (x) = \int \frac{d x}{\sqrt{(x - a) (b - x)}} ?

S o l u t i o n :

l e t t i n g x = a \cos^{2} q + b \sin^{2} q

w h e r e a < x < b, 0 < q < \frac{π}{2}

w e g e t {\begin{cases} x - a = (b - a) \sin^{2} q \\ b - x = (b - a) \cos^{2} q \end{cases}

a n d \tan^{2} q = \frac{x - a}{b - x} o r \tan q = \sqrt{\frac{x - a}{b - x}}

d x = 2 (b - a) \sin q \cos q d q

i m p l y i n g t h a t

φ (x) = \int \frac{d x}{\sqrt{(x - a) (b - x)}} = \int 2 d q

φ (x) = 2 q + c = 2 \arctan (\sqrt{\frac{x - a}{b - x}}) + c . ✓

Commented by john santu last updated on 19/Nov/20

n i c e

Answered by som(math1967) last updated on 19/Nov/20

let x - a = z^{2}

dx = 2 zdz

\int \frac{2 zdz}{z \sqrt{b - a - z^{2}}}

2 \int \frac{dz}{\sqrt{b - a - z^{2}}}

{2 \sin}^{- 1} \frac{z}{\sqrt{b - a}} + c

{2 \sin}^{- 1} (\sqrt{\frac{x - a}{b - a}}) + c

Answered by MJS_new last updated on 19/Nov/20

\int \frac{d x}{\sqrt{(x - a) (b - x)}} =

[t = \frac{\sqrt{x - a}}{\sqrt{b - x}} \to d x = \frac{2}{b - a} \sqrt{x - a} \sqrt{{(b - x)}^{3}} d t]

= 2 \int \frac{d t}{t^{2} + 1} = 2 \arctan t = 2 \arctan \frac{\sqrt{x - a}}{\sqrt{b - x}} + C

Commented by liberty last updated on 19/Nov/20

a m a z i n g \dots y o u r m e t h o d v e r y f a s t e s t

s i r

Commented by MJS_new last updated on 19/Nov/20

I^{'} m just an old integral lover \dots

Commented by liberty last updated on 19/Nov/20

h a h a h a h a \dots . g r e a t

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