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

Villaretq0

Villaretq0

Answered question

2022-06-16

Most general antiderivative involving sec x
I'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 x
Then 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 x
Then 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 + C
However, 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?

Answer & Explanation

arhaitategr

arhaitategr

Beginner2022-06-17Added 13 answers

Explanation:
No need to substitute. Remember that the derivative of sec(x) is sec(x) tan(x) [which is why your substitution isn't working], and the derivative of tan(x) is sec 2 ( x ).
Edit: I'll keep going so it'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

Extrakt04

Beginner2022-06-18Added 5 answers

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.

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