How to find the antiderivative of (e^x)/(1+e^(2x))?

jamestkfunghpff

jamestkfunghpff

Answered question

2023-02-27

How to find the antiderivative of e x 1 + e 2 x ?

Answer & Explanation

Abbigail Joyce

Abbigail Joyce

Beginner2023-02-28Added 6 answers

write e x dx as d ( e x ) , then we obtain
d ( e x ) 1 + ( e x ) 2
with the substitution y = e x , we get
d ( y ) 1 + y 2
which is equal to
arctan ( y ) + C
Now substitute back y = e x :
arctan ( e x ) + C
Audrina Donaldson

Audrina Donaldson

Beginner2023-03-01Added 6 answers

We need to find e x 1 + e 2 x d x = 1 1 + ( e x ) 2 e x d x
Now let u = e x and so taking the differential on both sides gives d u = e x d x . Now we substitute both of these equations into the integral to get
1 1 + u 2 d u
This is a standard integral which evaluates to arctan u . Substituting back for x we get a final answer:
arctan e x + c

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