Most general antiderivative involving sec xI'm stumped on how to get the most general antiderivative, F(x) , of f ( x ) = e x + 3 s e c x ( t a n x + s e c x ).First, I split the equation on addition, since ∫ [ f ( x ) + g ( x ) ] d x = ∫ f ( x ) d x + ∫ g ( x ) d x F ( x ) = ∫ e x d x + 3 ∫ s e c x ( t a n x + s e c x ) d xThen I use the Substitution Rule. sec(x) seems like a good choice for u since it's the inner function of a composition and du will encompass the other x terms u = s e c x ⇒ d u = ( t a n x + s e c x ) d xThen I have F ( x ) = e x + 3 ∫ ( u ) d u F ( x ) = e x + 3 u 2 2 + C F ( x ) = e x + 3 s e c 2 x 2 + CHowever, the answer key says the correct answer is F ( x ) = e x + 3 s e c x + 3 t a n x + C.Can someone please point out where I've gone wrong here?

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Most general antiderivative involving sec xI&#039;m stumped on how to get the most general antiderivative, F(x) , of   f  (  x  )  =      e    x    +  3  s  e  c  x  (  t  a  n  x  +  s  e  c  x  ).First, I split the equation on addition, since   ∫  [  f  (  x  )  +  g  (  x  )  ]  d  x  =  ∫  f  (  x  )  d  x  +  ∫  g  (  x  )  d  x  F  (  x  )  =  ∫      e    x    d  x  +  3  ∫  s  e  c  x  (  t  a  n  x  +  s  e  c  x  )  d  xThen I use the Substitution Rule. sec(x) seems like a good choice for u since it&#039;s the inner function of a composition and du will encompass the other x terms  u  =  s  e  c  x  ⇒  d  u  =  (  t  a  n  x  +  s  e  c  x  )  d  xThen I have   F  (  x  )  =      e    x    +  3  ∫  (  u  )  d  u    F  (  x  )  =      e    x    +            3              u        2              2    +  C    F  (  x  )  =      e    x    +            3      s      e              c        2            x        2    +  CHowever, the answer key says the correct answer is   F  (  x  )  =      e    x    +  3  s  e  c  x  +  3  t  a  n  x  +  C.Can someone please point out where I&#039;ve gone wrong here?

arhaitategr · Accepted Answer

Explanation:No need to substitute. Remember that the derivative of sec(x) is sec(x) tan(x) [which is why your substitution isn&#039;t working], and the derivative of tan(x) is       sec    2    ⁡  (  x  ).Edit: I&#039;ll keep going so it&#039;s clear. Beginning after your splitting the equation:  ∫      e    x    d  x     +     ∫  3  sec  ⁡  (  x  )  tan  ⁡  (  x  )  d  x     +     ∫  3      sec    2    ⁡  (  x  )  d  x

Extrakt04 · Accepted Answer

Explanation:From   u  =  sec  ⁡  x one rather gets   d  u  =  sec  ⁡  x  tan  ⁡  x  d  x, not   d  u  =  (  tan  ⁡  x  +  sec  ⁡  x  )  d  x, this is a first mistake.

Most general antiderivative involving sec x I'm stumped on how to get the most general antiderivati

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