Evaluate the following intregals: 1 3+2 + dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{3+2\sin\text{x}+\cos\text{x}}\ \text{dx}$

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Answer

Let $\text{I}=\int\frac{1}{3+2\sin\text{x}+\cos\text{x}}\ \text{dx}$
Putting $\sin\text{x}=\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}},\cos\text{x}=\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
$\text{I}=\int\frac{1}{3+2\Bigg(\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)+\Bigg(\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\Bigg)}$
$=\int\frac{\Big(1+\tan^2\frac{\text{x}}{2}\Big)}{3\Big(1+\tan^2\frac{\text{x}}{2}\Big)+4\tan\frac{\text{x}}{2}+1-\tan^2\frac{\text{x}}{2}}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{3+3\tan^2\frac{\text{x}}{2}+4\tan\frac{\text{x}}{2}+1-\tan^2\frac{\text{x}}{2}}\ \text{dx}$
$=\int\frac{\sec^2\frac{\text{x}}{2}}{2\tan^2\frac{\text{x}}{2}+4\tan\frac{\text{x}}{2}+4}\ \text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$\text{I}=\frac{2}{2}\int\frac{\text{dt}}{\text{t}^2+2\text{t}+2}$
$=\int\frac{\text{dt}}{\text{t}^2+2\text{t}+1-1+2}$
$\text{I}=\int\frac{\text{dt}}{(\text{t}+1)^2+(1)^2}$
$=\tan^{-1}(\text{t}+1)+\text{C}$
$\text{I}=\tan^{-1}\Big(\tan\frac{\text{x}}{2}+1\Big)+\text{C}$

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