=∫dx2+sin⁡x+cos⁡x=∫dx2+2tan⁡x/21+tan2⁡x/2+1−tan2⁡x/21+tan2⁡x/2=∫1+tan2⁡x/2tan2⁡x/2+2tan⁡x/2+3dx=∫1+tan2⁡x/2tan2⁡x/2+2tan⁡x/2+3dx Put tan⁡x2=t…12sec2⁡x2dx=dt∴ I=∫2dtt2+2t+3=∫dt(t+1)2+2=212tan⁡t+12+c=2tan⁡tan⁡x2+22+c

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$\int \frac{d x}{2 + \sin x + \cos x} .$

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$\sqrt{2} T a n \{\frac{T a n x / 2 + 2}{\sqrt{2}}\} + C$

$\sqrt{2} T a n^{- 1} \{\frac{T a n x / 2 + 2}{\sqrt{2}}\} + C$

$\sqrt{2} C o t \{\frac{C o t x / 2 + α}{\sqrt{2}}\} + C$

$2 T a n \{\frac{\frac{T a n x}{2} + 1}{\sqrt{2}}\} + C$

answer is B.

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Detailed Solution

$\begin{array}{l} = \int \frac{d x}{2 + \sin x + \cos x} \\ = \int \frac{d x}{2 + \frac{2 \tan x / 2}{1 + \tan^{2} x / 2} + \frac{1 - \tan^{2} x / 2}{1 + \tan^{2} x / 2}} \\ = \int \frac{1 + \tan^{2} x / 2}{\tan^{2} x / 2 + 2 \tan x / 2 + 3} d x \\ = \int \frac{1 + \tan^{2} x / 2}{\tan^{2} x / 2 + 2 \tan x / 2 + 3} d x \\ Put \tan \frac{x}{2} = t \dots \frac{1}{2} \sec^{2} \frac{x}{2} d x = d t \\ ∴ I = \int \frac{2 d t}{t^{2} + 2 t + 3} \\ = \int \frac{d t}{(t + 1)^{2} + 2} \\ = 2 \frac{1}{\sqrt{2}} \tan \frac{t + 1}{\sqrt{2}} + c \\ = \sqrt{2} \tan \frac{\tan \frac{x}{2} + 2}{\sqrt{2}} + c \end{array}$