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

Question

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

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Answer

Let $\text{I}=\int\frac{1}{2+\sin\text{x}+\cos\text{x}}\text{dx}$
Putting $\sin\text{x}=\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}},\cos\text{x}=\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}$
$\text{I}=-\int\frac{1}{2+\begin{pmatrix}\frac{2\tan\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\end{pmatrix}+\begin{pmatrix}\frac{1-\tan^2\frac{\text{x}}{2}}{1+\tan^2\frac{\text{x}}{2}}\end{pmatrix}}\text{dx}$
$=\int\frac{\begin{pmatrix}1+\tan^2\frac{\text{x}}{2}\end{pmatrix}}{2+2\tan^2\frac{\text{x}}{2}+2\tan\frac{\text{x}}{2}+1-\tan^2\frac{\text{x}}{2}}\text{dx}$
$=\int\frac{\begin{pmatrix}\sec^2\frac{\text{x}}{2}\end{pmatrix}}{\tan^2\frac{\text{x}}{2}+\tan\frac{\text{x}}{2}+3}\text{dx}$
Let $\tan\frac{\text{x}}{2}=\text{t}$
$\frac{1}{2}\sec^2\frac{\text{x}}{2}\text{dx}=\text{dt}$
$\text{I}=\int\frac{2\text{dt}}{\text{t}^2+2\text{t}+3}$
$\text{I}=2\int\frac{\text{dt}}{\text{t}^2+2\text{t}+1-1+3}$
$\text{I}=2\int\frac{\text{dt}}{(\text{t}+1)^2+(\sqrt2)^2}$
$=\frac{2}{\sqrt2}\tan^{-1}\Big(\frac{\text{t}+1}{\sqrt2}\big)+\text{C}$
$\text{I}=\sqrt{2}\tan^{-1}\Big(\frac{\tan\frac{\text{x}}{2}+1}{\sqrt2}\Big)+\text{C}$

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