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$\int \frac{dx}{\cos^{3} x \sqrt{2 \sin 2 x}}$ is equal to

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a

tan⁡x+tan5/2⁡x5+C

b

tan⁡x+25tan5/2⁡x+C

c

2tan⁡x+25tan5/2⁡x+C

d

None of the above

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detailed solution

Correct option is A

Let I=∫dxcos3⁡x2sin⁡2x=∫dxcos3⁡x4sin⁡xcos⁡x=12∫dxcos7/2⁡xsin1/2⁡x=12∫sec4⁡xtan⁡xdx=12∫1+tan2⁡xsec2⁡xtan⁡xdx Put tan⁡x=t⇒sec2⁡xdx=dt∴ I=12∫1+t2tdt =12∫t−1/2dt+12∫t3/2dt =t1/2+t5/25+C =tan⁡x+15tan5/2⁡x+C

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