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:

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

##### 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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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