If J=∫sec3⁡xdx then J is equal to

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|]

If J=∫sec3⁡xdx then J is equal to

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

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$(\sec x \tan x) + \frac{1}{2} \log |\sec x + \tan x| + C$

$\sec x \tan x - \log |\sec x + \tan x| + C$

$\frac{1}{2} \sec x \tan x + \log |\sec x + \tan x| + C$

$\frac{1}{2} [s e c x \tan x + \log |\sec x + \tan x|] + C$

answer is D.

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Detailed Solution

$Let J = \int \sec^{3} x d x$

$J = \int \sec x \sec^{2} x d x$

Using by parts

$\begin{array}{l} J = \sec x \int \sec^{2} d x - \int [\frac{d}{d x} (\sec x) \int \sec^{2} x d x] d x \\ J = \sec x \tan x - \int \sec x \tan x \tan x d x \\ J = \sec x \tan x - \int \sec x (\sec^{2} x - 1) d x \\ 2 J = \sec x \tan x + \log | \sec x + \tan x | \\ J = \frac{1}{2} [\sec x \tan x + \log | \sec x + \tan x |] \end{array}$