JEE Mains · Maths · STD 12 - 7.1 indefinite integral
The integral \(\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x\) is equal to
- A \(\frac{1}{2} \log _{ c }\left|\frac{\tan \left(\frac{ x }{2}+\frac{\pi}{12}\right)}{\left(\frac{ x }{2}+\frac{\pi}{6}\right)}\right|+ C\)
- B \(\frac{1}{2} \log _{ e }\left|\frac{\tan \left(\frac{ x }{2}+\frac{\pi}{6}\right)}{\left(\frac{ x }{2}+\frac{\pi}{3}\right)}\right|+ C\)
- C \(\log _{ c }\left|\frac{\tan \left(\frac{ x }{2}+\frac{\pi}{6}\right)}{\tan \left(\frac{ x }{2}+\frac{\pi}{12}\right)}\right|+ C\)
- D \(\frac{1}{2} \log _{ c }\left|\frac{\tan \left(\frac{ x }{2}-\frac{\pi}{12}\right)}{\tan \left(\frac{ x }{2}-\frac{\pi}{6}\right)}\right|+C\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{2} \log _{ c }\left|\frac{\tan \left(\frac{ x }{2}+\frac{\pi}{12}\right)}{\left(\frac{ x }{2}+\frac{\pi}{6}\right)}\right|+ C\)
Step-by-step Solution
Detailed explanation
\(I=\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x\) \(\frac{\sqrt{3}}{2} \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(\frac{\sqrt{3}}{2}+\sin 2 x\right)} d x\)…
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