Evaluate the following intregals: 1 3 + dx - Vidyadip

Question

Evaluate the following intregals: $\int\frac{1}{\sqrt{3}\sin\text{x}+\cos\text{x}}\ \text{dx}$

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Answer

Let $\text{I}=\int\frac{1}{\sqrt{3}\sin\text{x}+\cos\text{x}}\ \text{dx}$
Let $\sqrt{3}=\text{r}\cos\theta,\text{and }1=\text{r}\sin\theta$
$\tan\theta=\frac{1}{\sqrt{3}}$
$\theta=\frac{\pi}{6}$
$\text{r}=\sqrt{3+1}=2$
$\text{I}=\int\frac{1}{\text{r}\cos\theta\sin\text{x}+\text{r}\sin\theta\cos\text{x}}\ \text{dx}$
$=\frac{1}{\text{r}}\int\frac{1}{\sin(\text{x}+\theta)}\text{dx}$
$=\frac{1}{2}\int\text{cosec}(\text{x}+\theta)\text{dx}$
$=\frac{1}{2}\log\Big|\tan\Big(\frac{\text{x}}{2}+\frac{\theta}{2}\Big)\Big|+\text{c}$
$\text{I}=\frac{1}{2}\log\Big|\tan\Big(\frac{\text{x}}{2}+\frac{\pi}{12}\Big)\Big|+\text{C}$

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