Question
Evaluate: $\int \frac{1}{\sin x+\sqrt{3} \cos x} d x$
✓
Answer
$(a) :$ Let $I=\int \frac{1}{\sin x+\sqrt{3} \cos x} d x$
$=\frac{1}{2} \int \frac{d x}{\frac{1}{2} \sin x+\frac{\sqrt{3}}{2} \cos x}$
$\Rightarrow I=\frac{1}{2} \int \frac{1}{\sin \left(x+\frac{\pi}{3}\right)} d x=\frac{1}{2} \int \operatorname{cosec}\left(x+\frac{\pi}{3}\right) d x$
$\Rightarrow I=\frac{1}{2} \log \left|\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right|+C$
${\left[\because \int \operatorname{cosec} x d x=\log \left|\tan \frac{x}{2}\right|+C\right]}$
$=\frac{1}{2} \int \frac{d x}{\frac{1}{2} \sin x+\frac{\sqrt{3}}{2} \cos x}$
$\Rightarrow I=\frac{1}{2} \int \frac{1}{\sin \left(x+\frac{\pi}{3}\right)} d x=\frac{1}{2} \int \operatorname{cosec}\left(x+\frac{\pi}{3}\right) d x$
$\Rightarrow I=\frac{1}{2} \log \left|\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right|+C$
${\left[\because \int \operatorname{cosec} x d x=\log \left|\tan \frac{x}{2}\right|+C\right]}$
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