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JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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The value of ∫x−ab−xdx\int \sqrt{\frac{x-a}{b-x}} d x is equal to
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Correct option: B
(a−b)[sin⁡2θ2+θ]+c(a-b)\left[\frac{\sin 2 \theta}{2}+\theta\right]+c, where asin⁡2θ+bcos⁡2θ=xa \sin ^2 \theta+b \cos ^2 \theta=x
The correct answer is A because the substitution x=asin⁡2θ+bcos⁡2θx = a\sin^2 \theta + b\cos^2 \theta simplifies the integral to a form that can be easily integrated, resulting in (a−b)(sin⁡2θ2+θ)+c(a-b)(\frac{\sin 2\theta}{2} + \theta) + c.

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