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Perpendicular Distance of a Point from a Line

If ∫dx/√x -...

Question

If $\int \frac{d x}{\sqrt{(x - α) (β - x)}} = A {tan}^{- 1} f (x) + C$ then

A

A = 1 / 2

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B

f (x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x - (\frac{α + β}{2})}{\sqrt{(x - α) (β - x)}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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C

f (x) = \sqrt{(x - α) (β - x)}

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D

A = 1

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Solution

The correct options are
A $f (x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x - (\frac{α + β}{2})}{\sqrt{(x - α) (β - x)}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
D $A = 1$
Given : $\int \frac{d x}{\sqrt{(x - α) (β - x)}} = A {tan}^{- 1} f (x) + C$
$I = \int \frac{d x}{\sqrt{(x - α) (β - x)}} = \int \frac{d x}{\sqrt{\frac{(α + β)}{4} - α β - {[x - (\frac{α + β}{2})]}^{2}}}$

$I = \int \frac{d x}{\sqrt{\frac{(α + β)}{4} - {[x - (\frac{α + β}{2})]}^{2}}} = {sin}^{- 1} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x - (\frac{α + β}{2})}{\frac{α - β}{2}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
$

$I = {tan}^{- 1} \frac{x - (\frac{α + β}{2})}{\sqrt{\frac{(α + β)}{4} - α β - {[x - (\frac{α + β}{2})]}^{2}}}$

$I = {tan}^{- 1} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x - (\frac{α + β}{2})}{\sqrt{- x^{2} + (α + β) x - α β}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = {tan}^{- 1} ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x - (\frac{α + β}{2})}{\sqrt{(x - α) (β - x)}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ + C$

$A = 1 f (x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{x - (\frac{α + β}{2})}{\sqrt{(x - α) (β - x)}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

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