MHT CET · Maths · Indefinite Integration
\(\int \frac{1}{\cos x+\sqrt{3} \sin x} d x=\)
- A \(2 \log \left[\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)\right]+\mathrm{c}\)
- B \(\frac{1}{2} \log \left[\tan \left(\frac{x}{2}-\frac{\pi}{12}\right)\right]+c\)
- C \(\frac{1}{2} \log \left[\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)\right]+\mathrm{c}\)
- D \(2 \log \left[\tan \left(\frac{x}{2}-\frac{\pi}{12}\right)\right]+c\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{2} \log \left[\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)\right]+\mathrm{c}\)
Step-by-step Solution
Detailed explanation
Dividing numerator and denominator by 2 , we get
\(=\frac{1}{2} \int \frac{\mathrm{dx}}{\left(\frac{1}{2} \cos \mathrm{x}+\frac{\sqrt{3}}{2} \sin \mathrm{x}\right)}=\frac{1}{2} \int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}+\frac{\pi}{6}\right)}=\frac{1}{2} \log\) \(\left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)\right|+\mathrm{C}\)
\(=\frac{1}{2} \int \frac{\mathrm{dx}}{\left(\frac{1}{2} \cos \mathrm{x}+\frac{\sqrt{3}}{2} \sin \mathrm{x}\right)}=\frac{1}{2} \int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}+\frac{\pi}{6}\right)}=\frac{1}{2} \log\) \(\left|\tan \left(\frac{\mathrm{x}}{2}+\frac{\pi}{12}\right)\right|+\mathrm{C}\)
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