Split Using Partial Fraction Decomposition 1/(x^2-1)

Problem

1/(x2−1)

Solution

1. Factor the denominator using the difference of squares formula a2−b2=(a−b)*(a+b)

1/(x2−1)=1/((x−1)*(x+1))

2. Set up the partial fraction decomposition with unknown constants A and B

1/((x−1)*(x+1))=A/(x−1)+B/(x+1)

3. Multiply both sides by the common denominator (x−1)*(x+1) to clear fractions.

1=A*(x+1)+B*(x−1)

4. Solve for A by substituting x=1 into the equation.

1=A*(1+1)+B*(1−1)

1=2*A

A=1/2

5. Solve for B by substituting x=−1 into the equation.

1=A*(−1+1)+B*(−1−1)

1=−2*B

B=−1/2

6. Substitute the values of A and B back into the decomposition form.

1/(x2−1)=(1/2)/(x−1)+(−1/2)/(x+1)

Final Answer

1/(x2−1)=1/(2*(x−1))−1/(2*(x+1))

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