If     \\ \int e^{\sec x}\left ( \sec x \tan x f(x) +\sec x \tan x +\sec^{2}x \right )dx=e^{\sec x}f(x)+C   , then a possible choice of f(x)  is  :

  • Option 1)

    \sec x + \tan x +\frac{1}{2}

  • Option 2)

    \sec x - \tan x -\frac{1}{2}

  • Option 3)

    \sec x + x \:\tan x -\frac{1}{2}

  • Option 4)

    x \sec x + \tan x +\frac{1}{2}

     

 

Answers (1)

\\ \int e^{\sec x}\left ( \sec x \tan x f(x) +\sec x \tan x +\sec^{2}x \right )dx\\\\\\\:=e^{\sec x}f(x)+C

differentiating both sides 

\\e^{\sec x}\left ( \sec x \tan x f(x) + \sec x \tan x + \sec^{2}x\right )\\\\\\\:=e^{\sec x}f^{'}(x)+e^{\sec x}\sec x \tan x f^{'}(x)\\\\\\\:\therefore \sec x \tan x +\sec ^{2}x=f'(x)\\\\\\\:f(x)=\sec x +\tan x+d


Option 1)

\sec x + \tan x +\frac{1}{2}

Option 2)

\sec x - \tan x -\frac{1}{2}

Option 3)

\sec x + x \:\tan x -\frac{1}{2}

Option 4)

x \sec x + \tan x +\frac{1}{2}

 

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