Mathematics

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:


ANSWER

$$\sec { x } +\tan { x } +\cfrac { 1 }{ 2 } $$


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

Differentiating both sides w,r.t $$x$$

$$ { e }^{ \sec { x }  }\left( \sec { x } \tan { x } f(x)+(\sec { x } \tan { x } +\sec ^{ 2 }{ x }  \right) ={ e }^{ \sec { x }  }.\sec { x } \tan { x } f(x)+{ e }^{ \sec { x }  }.f'(x)$$

$$f'(x)=\sec ^{ 2 }{ x } +\tan { x } \sec { x } \Rightarrow f(x)=\tan { x } +\sec { x } +c$$
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Single Correct Medium Published on 17th 09, 2020
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