First slide

Methods of integration

Question

$\int \sin^{- 1} \sqrt{\frac{x}{a + x}} d x$ is equal to

Moderate

A

$(x + a) \tan^{- 1} \sqrt{\frac{x}{a}} - \sqrt{a x} + C$
B

$(x + a) \tan^{- 1} \sqrt{\frac{x}{a}} + \sqrt{a x} + C$
C

$(x + a) \cot^{- 1} \sqrt{\frac{x}{a}} - \sqrt{a x} + C$
D

None of these

Solution

Put $x = a \tan^{2} θ = d x = 2 a \tan θ \sec^{2} θ d θ$
$∴ \int \sin^{- 1} \sqrt{\frac{x}{a + x}} = \int θ .2 a \tan θ \sec^{2} θ d θ$

   $= 2 a [θ . \frac{\tan^{2} θ}{2} - \int \frac{\tan^{2} θ}{2} d θ]$

   $= a θ \tan^{2} θ - a \int (\sec^{2} θ - 1) + C$

   $= a θ (1 + \tan θ) - a \tan θ + C$

   $= (x + a) \tan^{- 1} \sqrt{\frac{x}{a}} - \sqrt{a x} + C .$

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