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

Problem

1/(x*(x2+1))

Solution

1. Set up the partial fraction decomposition by identifying the factors in the denominator. Since x is a linear factor and x2+1 is an irreducible quadratic factor, the form is:

1/(x*(x2+1))=A/x+(B*x+C)/(x2+1)

2. Multiply both sides by the common denominator x*(x2+1) to clear the fractions:

1=A*(x2+1)+(B*x+C)*x

3. Expand the right side of the equation:

1=A*x2+A+B*x2+C*x

4. Group the terms by the powers of x

1=(A+B)*x2+C*x+A

5. Equate the coefficients from both sides to create a system of equations:

A+B=0

C=0

A=1

6. Solve for the remaining constant B using the value of A

1+B=0⇒B=−1

7. Substitute the values A=1 B=−1 and C=0 back into the decomposition form:

1/(x*(x2+1))=1/x+(−1*x+0)/(x2+1)

Final Answer

1/(x*(x2+1))=1/x−x/(x2+1)

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