COMEDK · Maths · 28. Indefinite Integration
\(\int \sqrt{\operatorname{cosec} x-1} d x=\)
- A \(\log \left|\sin x+\dfrac{1}{2}+\sqrt{\sin ^2 x+\sin x}\right|+c\)
- B \(\log \left|\sin x+\dfrac{1}{2}+\sqrt{\sin ^2 x+\dfrac{1}{2}+\sin x}\right|+c\)
- C \(\log \left|\sin x+1+2 \sqrt{\sin ^2 x+\sin x}\right|+c\)
- D \(\log \left|\sin x+\sqrt{\sin ^2 x+\sin x}\right|+c\)
Answer & Solution
Correct Answer
(A) \(\log \left|\sin x+\dfrac{1}{2}+\sqrt{\sin ^2 x+\sin x}\right|+c\)
Step-by-step Solution
Detailed explanation
Let \(I = \int \sqrt{\operatorname{cosec} x - 1} dx = \int \sqrt{\dfrac{1}{\sin x} - 1} dx = \int \sqrt{\dfrac{1 - \sin x}{\sin x}} dx\). Multiply numerator and denominator by \(\sqrt{1 + \sin x}\):…
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