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Question: If I = \(\int_{}^{}\frac{dx}{\sqrt{(x - \alpha)(\beta - x)}}\). (b \<a) then value of I is:...

If I = dx(xα)(βx)\int_{}^{}\frac{dx}{\sqrt{(x - \alpha)(\beta - x)}}. (b <a) then value of I is:

A

sin–1(2xαββα)\left( \frac{2x - \alpha - \beta}{\beta - \alpha} \right)+ C

B

sin–1(xαββα)\left( \frac{x - \alpha - \beta}{\beta - \alpha} \right)

C

sin (2x+βαα+β)\left( \frac{2x + \beta - \alpha}{\alpha + \beta} \right)

D

None of these

Answer

sin–1(2xαββα)\left( \frac{2x - \alpha - \beta}{\beta - \alpha} \right)+ C

Explanation

Solution

Put t = 12\frac{1}{2} (x – a + x – b)

= x – 12\frac{1}{2} (a + b), so that

(x – a) (b – x) =[β12(α+β)t]\left\lbrack \beta - \frac{1}{2}(\alpha + \beta) - t \right\rbrack

= 14\frac{1}{4} (b – a)2 – t2

\ I = sin–1 (2tβα)\left( \frac{2t}{\beta - \alpha} \right)+ C = sin–1 (2xαββα)\left( \frac{2x - \alpha - \beta}{\beta - \alpha} \right)+ C