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$If J = \int \sec^{3} x d x then J is equal to$

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a

secxtanx+12logsecx+tanx+C

b

secxtanx−logsecx+tanx+C

c

12secxtanx+logsecx+tanx+C

d

12secxtanx+logsecx+tanx+C

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detailed solution

Correct option is D

Let J=∫sec3⁡xdxJ=∫secxsec2xdxUsing by partsJ=sec⁡x∫sec2⁡dx−∫ddx(sec⁡x)∫sec2⁡xdxdxJ=sec⁡xtan⁡x−∫sec⁡xtan⁡xtan⁡xdxJ=sec⁡xtan⁡x−∫sec⁡xsec2⁡x−1dx2J=sec⁡xtan⁡x+log⁡|sec⁡x+tan⁡x|J=12[sec⁡xtan⁡x+log⁡|sec⁡x+tan⁡x|]

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