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$\int \frac{\sin^{3} x d x}{(\cos^{4} x + 3 \cos^{2} x + 1) \tan^{- 1} (\sec x + \cos x)}$

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a

logTan−1Secx+Cosx+C

b

logCot−1Cosex+Cotx+C

c

logTan−1Secx−Tanx+C

d

logCot−1Cosex−Cotx+C

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detailed solution

Correct option is A

I=∫sin3⁡xdxcos4⁡x+3cos2⁡x+1tan−1⁡(sec⁡x+cos⁡x) Put tan−1⁡(sec⁡x+cos⁡x)=t or 11+(sec⁡x+cos⁡x)2(sec⁡xtan⁡x−sin⁡x)dx=dt or cos2⁡xcos2⁡x+1+cos2⁡x2sin⁡x1cos2⁡x−1dx=dtor cos2⁡xsin⁡x1−cos2⁡xcos2⁡xcos2⁡x+1+cos4⁡x+2cos2⁡xdx=dtor sin3⁡xcos4⁡x+3+cos2⁡x+1dx=dtI=∫dtt=log⁡|t|+c =log⁡tan−1⁡(sec⁡x+cos⁡x)+c

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