How do you use partial fraction decomposition to decompose the fraction to integrate #(2x)/(1-x^3)#?

Answer 1

#int(2x)/(1-x^3)dx=-2/3ln|x-1|+1/3ln|1+x+x^2|-3[2/sqrt3tan^(-1)((2x+1)/sqrt3)]#

Factors of #1-x^3# are #(1-x)(1+x+x^2)#, hence partial fractions of #(2x)/(1-x^3)# are of the form
#(2x)/(1-x^3)=A/(1-x)+(Bx+C)/(1+x+x^2)#
i.e. #(2x)/(1-x^3)=(A(1+x+x^2)+(Bx+C)(1-x))/(1-x^3)#
Comparing coefficients of constant term, #x# and #x^2# in numerator
#A+C=0#, #A+B-C=2# and #A-B=0#
solving these we get #A=B=2/3#, #C=-2/3# and hence
#(2x)/(1-x^3)=2/(3(1-x))+(2x-2)/(3(1+x+x^2))# and
#int(2x)/(1-x^3)dx=int2/(3(1-x))dx+int(2x-2)/(3(1+x+x^2))dx#
Now #int2/(3(1-x))dx=2/3int1/(1-x)dx=-2/3ln|x-1|#
and #int(2x-2)/(3(1+x+x^2))dx#
= #int(2x+1)/(3(1+x+x^2))dx-int3/(1+x+x^2)dx#
= #1/3ln|1+x+x^2|-3[2/sqrt3tan^(-1)((2x+1)/sqrt3)]#
For former observe #d/(dx)(1+x+x^2)=2x+1# and for latter, we can have
#int3/(1+x+x^2)dx=3int1/((x+1/2)^2+3/4)dx=3int1/((x+1/2)^2+(sqrt3/2)^2)dx#
= #3[2/sqrt3tan^(-1)((2x+1)/sqrt3)]#
Hence #int(2x)/(1-x^3)dx=-2/3ln|x-1|+1/3ln|1+x+x^2|-3[2/sqrt3tan^(-1)((2x+1)/sqrt3)]#
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Answer 2

To decompose the fraction (2x)/(1-x^3) using partial fraction decomposition, follow these steps:

  1. Factor the denominator: 1 - x^3 = (1 - x)(1 + x + x^2).
  2. Write the fraction in the form of partial fractions: (2x)/(1-x^3) = A/(1 - x) + (Bx + C)/(1 + x + x^2).
  3. Multiply both sides by the denominator of the original fraction to clear the fractions.
  4. Equate coefficients of like terms.
  5. Solve for the unknown coefficients A, B, and C.
  6. Integrate each partial fraction separately.
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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.

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