How do you integrate #int (1) / (x * ( x^2 - 1 )^2)# using partial fractions?

Answer 1

The answer is #=-1/2ln(|x^2-1|)+ln(|x|)-1/(2(x^2-1))+C#

Reminder

#x^2-1=(x+1)(x-1)#

Now let's break it down into partial fractions.

#1/(x(x^2-1)^2)=1/(x(x+1)^2(x-1)^2)#
#=A/x+B/(x+1)^2+C/(x+1)+D/(x-1)^2+E/(x-1)#
#=(A(x+1)^2(x-1)^2+B(x)(x-1)^2+C(x)(x-1)^2(x+1)+D(x)(x+1)^2+E(x)(x+1)^2(x-1))/(x(x^2-1)^2)#

Compare the numerators; the denominators are the same.

#1=A(x+1)^2(x-1)^2+B(x)(x-1)^2+C(x)(x-1)^2(x+1)+D(x)(x+1)^2+E(x)(x+1)^2(x-1)#
Let #x=0#, #=>#, #1=A#
Let #x=1#, #=>#, #1=4D#, #=>#, #D=1/4#
Let #x=-1#, #=>#, #1=-4B#, #=>#, #B=-1/4#
Coefficients of #x^4#
#0=A+C+E#, #=>#, #C+E=-A=-1#
Coefficients of #x^3#
#0=B+D-C+E#, #=>#, #B+D=C-E=0#
#C=E=-1/2#

Consequently,

#1/(x(x^2-1)^2)=1/x+(-1/4)/(x+1)^2+(-1/2)/(x+1)+(1/4)/(x-1)^2+(-1/2)/(x-1)#

So,

#int(dx)/(x(x^2-1)^2)=int(dx)/x-int(1/4dx)/(x+1)^2-int(1/2dx)/(x+1)+int(1/4dx)/(x-1)^2-int(1/2dx)/(x-1)#
#=ln(|x|)+1/(4(x+1))-1/2ln(|x+1|)-1/(4(x-1))-1/2ln(|x-1|)+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 integrate ( \frac{1}{x(x^2 - 1)^2} ) using partial fractions, first factor the denominator completely: ( x(x^2 - 1)^2 = x(x - 1)^2(x + 1)^2 ).

Then, express the fraction as a sum of partial fractions:

[ \frac{1}{x(x^2 - 1)^2} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{(x - 1)^2} + \frac{D}{x + 1} + \frac{E}{(x + 1)^2} ]

Now, find the values of ( A ), ( B ), ( C ), ( D ), and ( E ) by finding a common denominator and equating coefficients. Then, integrate each partial fraction separately.

After integrating, the final answer will be in terms of ( \ln |x| ), ( \ln |x - 1| ), ( \frac{1}{x - 1} ), ( \ln |x + 1| ), and ( \frac{1}{x + 1} ), along with constants of integration.

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