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The value of $\int \frac{1 + \sin x}{1 - \sin x} d x$ is

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a

2tan⁡x2+π4+C

b

2tan⁡x2+π4+x+C

c

2tan⁡x2+π4−x+C

d

2tan2⁡x2+π4−x+C

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detailed solution

Correct option is C

∫1+sin⁡x1−sin⁡xdx=∫(1+sin⁡x)21−sin2⁡xdx=∫(sec⁡x+tan⁡x)2dx=∫sec2⁡x+∫sec2⁡x−1dx+2∫sec⁡xtan⁡xdx=2tan⁡x−x+2sec⁡x+C=21+sin⁡xcos⁡x−x+C=21−cos⁡x+π2sin⁡x+π2−x+C=22sin2⁡x2+π42sin⁡x2+π4cos⁡x2+π4−x+C=2tan⁡x2+π4−x+C

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