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$\int \frac{1}{1 + \sin x + \cos x} d x$

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a

12log⁡tan⁡x2+1+C

b

log⁡tan⁡(x+1)+C

c

log⁡tan⁡x2+1+C

d

None of these

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detailed solution

Correct option is C

Putting sin⁡x=2tan⁡x/21+tan2⁡x/2 and cos⁡x=1−tan2⁡x/21+tan2⁡x/2 , I=∫11+sin⁡x+cos⁡xdx=∫11+2tan⁡x/21+tan2⁡x/2+1−tan2⁡x/21+tan2⁡x/2dx=∫1+tan2⁡x/21+tan2⁡x/2+2tan⁡x/2+1−tan2⁡x/2dx=∫sec2⁡x/22+2tan⁡x/2dx Putting tan⁡x2=t and 12sec2⁡x2dx=dt or sec2⁡x2dx=2dt, we get I=∫2dt2+2t=∫1t+1dt=log⁡|t+1|+C=log⁡tan⁡x2+1+C

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