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Questions

$\int (\sqrt{\tan x} + \sqrt{\cot x}) dx$ is equal to

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a

2sin−1⁡(sin⁡x−cos⁡x)+C

b

2sin−1⁡(sin⁡x+cos⁡x)+C

c

2tan−1⁡(sin⁡x−cos⁡x)+C

d

None of the above

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detailed solution

Correct option is A

∫(tan⁡x+cot⁡x)dx=∫sin⁡xcos⁡x+cos⁡xsin⁡xdx=∫sin⁡x+cos⁡xsin⁡xcos⁡xdx=∫2(sin⁡x+cos⁡x)2sin⁡xcos⁡xdx=2∫sin⁡x+cos⁡x1−(sin⁡x−cos⁡x)2dxPut ⁡sin⁡x−cos⁡x=t⇒(cos⁡x+sin⁡x)dx=dt=2∫dt1−t2=2sin−1⁡t+C=2sin−1⁡(sin⁡x−cos⁡x)+C

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