Evaluate : 1 x-3 x d x - Vidyadip

Question

Evaluate : $\int \frac{1}{\sin x-\sqrt{3} \cos x} \cdot d x$

✓

Answer

For any two positive numbers $a$ and $b$, we can find an angle $\theta$, such that
$
\therefore \quad \sin \theta=\frac{a}{\sqrt{a^2-b^2}} \text { and } \cos \theta=\frac{b}{\sqrt{a^2-b^2}}
$
Using this we express $\sin x-\sqrt{3} \cos x$
$
\begin{aligned}
& =\sqrt{1+3}(\cos \theta \cdot \sin x-\sin \theta \cdot \cos x) \\
& =2 \cdot \sin (x-\theta) \\
& =2 \cdot \sin \left(x-\frac{\pi}{3}\right) \\
\therefore I & =\int \frac{1}{2 \cdot \sin \left(x-\frac{\pi}{3}\right)} \cdot d x \\
& =\frac{1}{2} \cdot \int \operatorname{cosec}\left(x-\frac{\pi}{3}\right) \cdot d x \\
& =\frac{1}{2} \cdot \log \left(\operatorname{cosec}\left(x-\frac{\pi}{3}\right)-\cot \left(x-\frac{\pi}{3}\right)\right)+c \\
& =\frac{1}{2} \cdot \log \left(\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)\right)+c
\end{aligned}
$

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