How do you use partial fraction decomposition to decompose the fraction to integrate #1/(1+e^x) #?

Answer 1

#int 1/(1+e^(x))\ dx=x-ln(1+e^(x))+C#.

This is not really a partial fraction problem, but instead it's just a trick. Write:

#1/(1+e^(x))=(1+e^(x)-e^(x))/(1+e^(x))#
#=(1+e^(x))/(1+e^(x))-e^(x)/(1+e^(x))=1-e^(x)/(1+e^(x))#

This means

#int 1/(1+e^(x))\ dx=int (1-e^(x)/(1+e^(x)))\ dx#
#=x-int e^(x)/(1+e^(x))\ dx#.
For this last integral, let #u=1+e^(x)# so that #du=e^(x)\ dx# and we get
#int e^(x)/(1+e^(x))\ dx=int\ (du)/u=ln|u|+C=ln(1+e^(x))+C# (note that #1+e^(x)>0# for all #x#).

Therefore,

#int 1/(1+e^(x))\ dx=x-ln(1+e^(x))+C#.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To decompose the fraction 1/(1+e^x) using partial fraction decomposition, first, express it as a sum of simpler fractions. Start by assuming that 1/(1+e^x) can be expressed as A/(1+e^x) + B/(1+e^(-x)). Then, find the values of A and B by equating the numerators of the original expression and the decomposed expression. Finally, integrate each simpler fraction individually to obtain the integral of the original expression.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7